Faster Isomorphism for Groups of Class 2

Abstract

The finite-group isomorphism problem asks whether two groups, given by their Cayley tables, are isomorphic. Despite decades of study, obtaining an $N^{o(\log N)}$-time algorithm for general groups of order $N$ remains a major open challenge. Groups of nilpotency class $2$ have long been a central obstacle to achieving this goal. Recent breakthroughs have established $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ with odd $p$ (Sun, STOC '23; Ivanyos-Mendoza-Qiao-Sun-Zhang, FOCS '24; Grochow-Qiao-Stange-Sun, STOC '25) and for groups of Frattini class $2$ (Grochow-Ivanyos-Qiao-Sun, FOCS '26). However, the problem remained open for general groups of class $2$. In this paper, we present an algorithm for testing isomorphism of groups of class $2$, with running time $N^{O((\log N{)}^{25/26}\log \log N)}$. Our algorithm combines a tensor-based restriction of the search space, an encoding of group relations by alternating matrix tuples, and an orbit computation to recover the remaining central power information.