I have two questions regarding to $p$-groups.

- A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of course there are many other $p$-groups with cyclic center that are not extraspecial. I would to know if there is any classification of $p$-groups with cyclic center.
- Is there any classification of $p$-groups of order $p^n$ and nilpotency class $k$ for suitable fixed $k$ and $n$?