Two groups $(G,\cdot)$ and $(H, \times)$ are said to be isomorphic iff there exists a homomorphism from $G$ to $H$ which is bijective. The group isomorphism problem is as follows: given two groups, check whether they are isomorphic or not. There are different ways to input a group, the two mostly used are by a Cayley table and by a generating set. Here I am assuming input groups are given by their Cayley table. More formally:
$\textbf{Group Isomorphism Problem}$
$\textbf{Input : }$ Two groups $(G,\cdot)$ and $(H,\times)$.
$\textbf{Decide : } $ Is $G \cong H$?
Let us assume that $n = |G| = |H|$
Group Isomorphism problem when input groups are given by Cayley table is not known to be in $\textbf{P}$ in general. Although there are group classes like the abelian group class for which the problem is known to be in polynomial time, groups which are the extension of an abelian group, simple groups etc. Even for nilpotent class two groups, no algorithm better than brute force is known.
A brute force algorithm for group isomorphism is given by Tarjan, which is as follows. Let $G$ and $H$ are two input groups, and let $S$ be a generating set of the group $G$. It is a well-known fact that every finite group admits a generating set of $\mathcal{O}(\log n)$ size and which can be found in polynomial time. The number of images of the generating set $S$ in the homomorphism from $G$ to $H$ is $n^{\log n}$ many. Now, check whether each possible homomorphism is bijective or not. The overall runtime will be $n^{\log n + \mathcal{O}(1)}$.
Let me first define the center of the group $G$:
$$Z(G) = \{g \in G \mid ag=ga, \forall a \in G\}$$
$Z(G)$ denotes the elements of the group $G$ which commutes with all other elements of the group $G$. Groups for which $G/Z(G)$ ( / used for quotient) is abelian are known as a nilpotent class two groups. To me it appears that nilpotent class two groups are the hardest instances to solve the group isomorphism problem. The meaning of "hardest instances" is: solving that case will allow researchers who work in group theory to solve the isomorphism problem of a large number of groups.
Initially, I thought that simple groups are the hardest instances as they are building blocks of all groups, but later came to know that the isomorphism problem for simple groups is in $\textbf{P}$.
Question: What is the hardest instance for the group isomorphism problem?