only slight improvements for the general case. In fact, Rosenbaum [@DBLP:journals/corr/abs-1304-3935] was able to improve it to $n^{1/2\log n+\mathcal{O}(1)}$. (See [@DBLP:journals/corr/Luks15] and [@DBLP:journals/tcs/RosenbaumW15] for related discussions on isomorphism of $p$-groups and solvable groups). For various classes of groups, better bounds are known (see further related work). However, even very limited classes of groups provide hard cases for isomorphism testing. One of the most prominent classes in this context is formed by the groups of prime exponent $p$ and nilpotency class $2$. Such groups possess a lot of extra structure, but despite this and despite a large body of research into this structure, even for this limited class, no better general bound has been proven. In fact, this class seems to be at the core of the problem. However, a formal reduction to this or a similar class is not known.

While there exists a vast collection of algebraic methods and heuristics for tackling the group isomorphism problem (see further related work), complexity theoretic and combinatorial aspects seem to be less developed. For example, in 2011, Timothy Gowers asked on Lipton’s blog [@gowers] whether there is an integer $m$ such that the isomorphism class of each finite group is determined by 