A finite $p$-group is said to be special if $Z(G)=G'$. Is there classification of special $p$-groups? (Please suggest references, if classification is done. If the classification is incomplete, please suggest the references, in which it is done for particular cases. The case $Z(G)=G'\cong \mathbb{Z}/p$ is very well known. )
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The standard definition of a special p-group is more restrictive than that. A $p$-group $G$ is special if either it is elementary abelian, or if $P'=Z(P)=\Phi(P)$ is elementary abelian. So in the second case both $Z(P)$ and $P/Z(P)$ are required to be elementary abelian.
I don't believe that there is any precise classification. One could reasonably conjecture that in some sense almost all finite groups are special 2-groups. Graham Higman's lower bound of $p^{2n^3/27 - O(n^2)}$ for the number of isomorphism classes of groups of order $p^n$ was obtained by estimating the number of special $p$-groups. (Sims later improved Higman's upper bound to $p^{2n^3/27 + O(n^{8/3})}$.)
answered Apr 17, 2013 at 8:11
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$\begingroup$ Is there a sharp upper bound, at least conjecturally ? $\endgroup$ Commented Apr 17, 2013 at 8:50
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2$\begingroup$ I haven't heard of such a conjecture. Wuth current techniques it seems that there will always be a nonconstant error term in the exponent. So we cannot really justify conjecturing that "almost all $p$-groups are special", although that statement is somehow true logarithmically. $\endgroup$ Commented Apr 17, 2013 at 9:13