computes the $k$-profile of groups. In particular, abelian groups are completely identified already by the least powerful of the algorithms.

The second aim of this paper is to understand when and how groups are characterized by their combinatorial properties. On the one hand this addresses the question whether combinatorial methods can solve the Group Isomorphism Problem. On the other hand it provides a way of quantifying similarity of non-isomorphic groups. Specifically, we construct pairs of arbitrarily large non-isomorphic groups that agree with respect to many isomorphism invariants but can still be distinguished with the $3$-dimensional WL-algorithm. More precisely these groups are of nilpotency class $2$ and prime exponent $p$. They are non-isomorphic but have the same $\Theta(\log n)$-profile. They also have highly similar commuting graphs.

For infinitely many $n$ there exist pairs of non-isomorphic groups order $n$ with bounded Weisfeiler-Leman dimension which

-   have equal $\Theta(\log n)$-profiles,

-   have commuting graphs that are indistinguishable for the $\mathcal{O}(\log(n))$-dimensional Weisfeiler-Leman algorithm (for graphs),

-   are of exponent $p$ and nilpotency class $2$, and

-   have equal sizes of conjugacy classes.

The theorem shows that the Weisfeiler-Leman algorithm can be more effective in distinguishing groups than in distinguishing graphs even when they are 