- $G = \{ x,y \mid x^{2^{n+1}} = 1, y^4 = 1, xy = yx^{ā1} \}$
- $H = \{ x,y \mid x^{2^{n+1}} = 1, y^4 = 1, xy = yx^{2^nā1}\}$
Can these groups of order $2^{n+3}$ and nilpotency class $n+1$ be distinguished by any reasonable (computable) set of invariants?
While for any set of reasonable invariants, there is surely a pair of groups that share the invariants, it is also surely true that for any pair of groups there is some invariant that distinguishes them.
I personally have not found any significant differences in the conjugacy classes, centralizers, proper subgroups, normalizers, normal subgroups, etc., but perhaps I overlooked something. I've been looking at simple groups lately, and have forgotten how muddy $p$-groups can be.