TITLE: Very generic question about Commutator and Center
QUESTION [10 upvotes]: Let $G$ be a finite group and $G'$ the commutator group of $G$.

What can I say about $G' \cap Z(G)$? 

Could you be as specific as possible about p-Groups?

REPLY [7 votes]: Well, this is sort of a broad question, but here a couple random facts.

$G'\cap Z(G)\leqslant \Phi(G)$ for all groups $G$.
Special $p$-groups are an interesting case of $p$-groups in which $G'\cap Z(G)= Z(G)=G'$.

There is also some kind of theorem about $[G:G'\cap Z(G)]^2$ dividing the degrees of a group's irreducible characters, which I believe I saw in Huppert, but I cannot remember the exact statement at the moment.