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


1. 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.
2. Is there any classification of $p$-groups of order $p^n$ and nilpotency class $k$ for suitable fixed $k$ and $n$?



classification of $p$-groups
Every $p$-group is a homomorphic image of a $p$-group with cyclic center of order $p$, so a classification (whatever that means) of $p$-groups with cyclic center would (more-or-less) yield a construction for all $p$-groups, and I would not hold my breath waiting for that. 


To see why a $p$-group $P$ is a homomorphic image of a $p$-group $G$ with center of order $p$, let $G$ be the regular wreath product of a cyclic group of order $p$ with $P$. Thus $G$ has an elementary abelian subgroup $E$ of order $p^{|P|}$, where $P$ permutes the cyclic factors of $E$ the way it permutes its own elements by right multiplication, and thus $P$ acts faithfully on $E$. Also, $G$ is the semidirect product of $E$ by $P$. It is easy to see that $E \cap {\bf Z}(G)$ has order $p$, so I need to show that every element of ${\bf Z}(G)$ lies in $E$. If $z \in {\bf Z}(G)$, write $z = au$, where $a \in E$ and $u \in P$. Since $z$ centralizes $E$ and $a$ centralizes $E$, it follows that $u$ centralizes $E$ and thus $u = 1$ by the faithfulness of the action. Thus $z = a \in E$, as required.


