Faster Isomorphism for Groups of Class 2

The finite-group isomorphism problem asks whether two groups, given by their Cayley tables, are isomorphic. Despite decades of study, obtaining an No(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 No(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.