Capable groups of order $32$ with GAP

Question

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. How can I find all capable groups $G$ of order 32 with $|Cent(G)|=10$, where $Cent(G)$ is the set of all centralizers of elements in the group $G$? Is it possible to find it with GAP?

Answer 1

In fact you can do this directly in $\mathsf{GAP}$. There is a function $\mathtt{Epicentre}$, and a group is capable if and only if its epicentre is trivial. It turns out that only small groups number $6$ and $18$ are capable from the candidates above.

Answer 2

A quick search in the small groups database shows that there are only 8 groups of order 32 having 10 centralizers: the groups indexed by 6, 7, 8, 18, 19, 20, 43, 44 (see here for more information). These are 2-groups of nilpotency class 3 respectively 4. There is a necessary condition for such groups to be capable as mentioned in this excellent post but this condition is never violated here, so we cannot exclude any from this list.

I don't know about further conditions but a brute force search in the small groups database shows that at least the groups indexed by 8 and 18 are capable:

SmallGroup(32,6) is the central quotient of SmallGroup(64,32)

SmallGroup(32,18) is the central quotient of SmallGroup(64,52)