Two p-groups of exponent $p$ with same number of conjugacy classes but non-isomorphic centers

Question

I am searching for an example of two $p$-groups of exponent $p$ with same number of conjugacy classes but not isomorphic centers.

Background: The corresponding unit groups of the centers of the group algebras over $\operatorname{GF}(p)$ are isomorphic but the center are not isomorphic for the original groups.

Answer 1

I have found some groups which satisfy your conditions, using the Computer Algebra System GAP.

In particular the groups in the GAP small group library with IDs [729,122] and [729,440] satisfy your requirements.

See below for the output from a GAP session:

gap> G:=SmallGroup(729,122);
<pc group of size 729 with 6 generators>
gap> H:=SmallGroup(729,440);
<pc group of size 729 with 6 generators>
gap> Exponent(G);
3
gap> Exponent(H);
3
gap> Length(ConjugacyClasses(G));
105
gap> Length(ConjugacyClasses(H));
105
gap> StructureDescription(Centre(G));
"C3 x C3 x C3"
gap> StructureDescription(Centre(H));
"C3 x C3"

I found these examples by searching the Small Groups library which is included in GAP, I filtered to get only the groups of exponent $p$ and then looked to see if I could find a pair with the same number of conjugacy classes but non isomorphic centres. You might find that you can find more examples in the same way. I stopped once I found one.

Once you have found the groups, you can use GAP to obtain a presentation for them. For example see this question.