In the building that once housed the City Duma (https://reveal.world/story/zdanie-gorodskoj-dumy-c-bashnej), and now is home to the Rimsky-Korsakov Music School, meetings of the Balakirev Circle—a fellowship of Russian composers founded by Mily Balakirev—were held in the 1860s. Later, thanks to an article by music critic Vladimir Stasov, the circle was renamed the "Mighty Handful": "How much poetry, feeling, talent, and skill there is in this small but already mighty handful of Russian musicians." The center of the musical and educational activities of this group of young composers was the Free Music School they opened, the successor of which is the modern educational institution.

The "Mighty Handful," the "New Russian Music School," or the Balakirev Circle. A community of Russian composers formed in the mid-19th century. The name "Mighty Handful" was popularized by the well-known music critic Vladimir Stasov—in Russia. In one of his articles, the critic noted: "How much poetry, feeling, talent, and skill there is in this small but already mighty handful of Russian musicians." The phrase became a catchphrase, and members of the musical community began to be called "the handful."
The composers of the "Mighty Handful" considered themselves heirs to the recently deceased Mikhail Glinka and dreamed of developing Russian national music. The spirit of democracy was in the air, and the Russian intelligentsia contemplated a cultural revolution without violence or bloodshed—exclusively through the power of art.
In Europe, the community of musicians was simply called the "Group of Five."
The first step toward the formation of the "Mighty Handful" was the arrival in 1855 in St. Petersburg of the gifted 18-year-old musician Mily Balakirev. With brilliant performances, the pianist attracted the attention not only of the discerning public but also of the most famous music critic of the time—Vladimir Stasov, who became the ideological inspirer of the composers' union. A year later, Balakirev met military engineer César Cui. In 1857, he met Modest Mussorgsky, a graduate of a military academy. In 1862, he met naval officer Nikolai Rimsky-Korsakov, and at the same time, common musical views were discovered with chemistry professor Alexander Borodin. Thus, the musical circle was formed. Balakirev introduced the novice musicians to the theory of composition, orchestration, and harmony. Together, the like-minded read Belinsky and Chernyshevsky, jointly opposed academic routine, and sought new forms—under the common idea of nationality as the main direction of music development.
Folk song as the basis for classical music. The handful collected folklore and studied Russian church singing. They organized entire musical expeditions. For example, Balakirev brought material from a trip along the Volga with poet Nikolai Shcherbina in 1860, which became the basis for an entire collection—"40 Russian Folk Songs."
Thanks to the efforts of the Balakirev circle, a free music school was opened for talented people of different social classes. The school hosted free concerts of works by the handful and like-minded composers. The school outlasted the Balakirev circle and operated until the revolution.
The 1870s separated the Balakirev circle. The "Mighty Handful" disbanded, but the five Russian composers continued to create. As Borodin wrote, individuality prevailed over the school, but "the common musical character, the common style inherent to the circle, remained": in the classrooms of the St. Petersburg Conservatory alongside Rimsky-Korsakov and in the work of successors—the Russian composers of the 20th century.

With the cessation of regular meetings of the five Russian composers, the growth, development, and living history of the "Mighty Handful" by no means ended. The center of the handful’s activity and ideology, mainly thanks to Rimsky-Korsakov’s pedagogical work, shifted to the classrooms of the St. Petersburg Conservatory, and starting from the mid-1880s, also to the "Belyayev Circle," where Rimsky-Korsakov was recognized as the head and leader for nearly 20 years. Then, at the beginning of the 20th century, he shared his leadership within a "triumvirate" with A. K. Lyadov, A. K. Glazunov, and a little later (from May 1907) N. V. Artsybushev. Thus, excluding Balakirev’s radicalism, the "Belyayev Circle" became a natural continuation of the "Mighty Handful."
Rimsky-Korsakov himself recalled this quite definitively in his "Chronicle of My Musical Life":
"Can the Belyayev Circle be considered a continuation of the Balakirev Circle? Was there a certain degree of similarity between the two, and what was the difference besides the change over time in its personal composition? The similarity, indicating that the Belyayev Circle is a continuation of the Balakirev Circle, besides the connecting links in the persons of myself and Lyadov, consisted in the common advanced and progressive nature of both; but the Balakirev Circle corresponded to a period of storm and pressure in the development of Russian music, while the Belyayev Circle was a period of calm progress; the Balakirev Circle was revolutionary, the Belyayev Circle was progressive..."
Sources:
https://www.classicalmusicnews.ru/articles/mighty-handful-10-facts/
https://ru.wikipedia.org/wiki/Могучая_кучка
https://adresaspb.com/news/peterburgskie-adresa-rimskogo-korsakova/