abelian groups has been studied in [@DBLP:journals/ijac/Gomaa10]. However, descriptive complexity of groups has been investigated considerably less than that of graphs. In contrast to this, the research body on the algorithmic Group Isomorphism Problem is extensive. The results can generally be divided into research with a more practical and research with a more theoretical focus.

*On the practical side* the best algorithms for isomorphism testing are typically implemented in computer algebra systems such as SAGE, MAGMA, GAP. Classical algorithms include the one by Smith [@smith_1996] (for solvable groups), the one by Eick, Leedham-Green and O’Brien (for $p$-groups) [@MR1904637; @MR1283739], as well as a general algorithm by Cannon and Holt [@DBLP:journals/jsc/CannonH03]. Newer algorithms have been developed by Wilson [@WilsonThesis] with numerous improvements over time together with Brooksbank and Maglione [@MR3591162]. More recent work introduces ever stronger invariants to distinguish groups quickly. We refer to [@DBLP:journals/corr/abs-1905-02518] for an overview and the most recent techniques and an algorithm incorporating many of them. Dietrich and Wilson report that current isomorphism tests are already infeasible in practice on some groups with orders in the thousands [@1806.08872].

In any case, in our work we focus on the *theoretical side*. As mentioned before, the best bound for 