I am studying $AC$-groups, i.e. groups in which the centralizer of every 
non-identity element is abelian. Now I need an example of a group in which
the centralizer of every non-identity element is non-abelian.
Where can I find and how can I classify them? 



Example of a group in which centralizers of every element are non-abelian
The smallest groups in which the centralizer of every element is non-abelian
have order $32$. You can find them with [GAP](http://www.gap-system.org) as follows:



```
gap> IsExample := G -> ForAll(ConjugacyClasses(G),
>                          cl->not IsAbelian(Centralizer(G,Representative(cl))));;
gap> n := 1;;
gap> repeat n := n + 1; sol := Filtered(AllGroups(n),IsExample); until sol <> [];
gap> sol;
[ <pc group of size 32 with 5 generators>, <pc group of size 32 with 5 generators> ]
gap> List(sol,IdGroup);
[ [ 32, 49 ], [ 32, 50 ] ]
gap> List(sol,StructureDescription);
[ "(C2 x D8) : C2", "(C2 x Q8) : C2" ]

```

