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


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*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 $n$? 

 A: This was mentioned as a brief comment by Derek Holt, but I think it deserves to be an answer.  For a group of order $p^n$ and class $c$, the coclass is $n-c$.  The known theory of classification by coclass is much richer than the known theory of classification by class; there is a nice account at http://www.ma.rhul.ac.uk/sepgm/Eick_Classification.pdf.  Part of the story is that one can also define coclass for infinite pro-$p$-groups, which is useful for the classification.  The isomorphism types of finite $p$-groups of fixed coclass can be assembled into a tree in a natural way, and the infinite paths from the root of the tree correspond to isomorphism classes of pro-$p$-groups.  
A: To echo Marty's sentiment, a classification of $p$-groups based on nilpotency class and size seems far off. As an example of where this problem can get complicated, see
http://www.degruyter.com/view/j/jgth.2011.14.issue-6/jgt.2010.081/jgt.2010.081.xml.
Basically, Halasi and Palfy construct a collection of $p$-groups (allowing $p$ to vary) of nilpotency class 2 where the number of conjugacy classes is not polynomial in the prime $p$. This is related to the (unsolved) question of whether the number of conjugacy classes of $UT_n(\mathbb{F}_q)$, the group of unipotent upper-triangular matrices over the field with $q$ elements, is polynomial in $q$.
To summarize, $p$-groups, even those of nilpotency class 2, get complicated.
A: 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.
A: I think that a ``proper" question which must be proposed instead of Question 1, is the following:
Is there a classification of finite $p$-groups $G$ such that both $G'$ and $Z(G)$ are cyclic?
The latter question is  inspired by the example of extra-special groups as in this case both center and commutator subgroups are cyclic with further conditions.
The answer of the above question is yes: To start finding a series of papers answering the question you may see:
A. A. Finogenov, Finite $p$-Groups With Cyclic Commutator Subgroup and Cyclic Center, Matematicheskie Zametki, Vol. 63, No. 6, pp. 911-922, June, 1998.
A: Classification of $p$-groups by nilpotency class is hard in general. As pointed above, coclass seems to be a better invariant. On the other hand, Ahmad, Magidin and Morse recently finished a classification of 2-generator $p$-groups of class 2:
Ahmad, Azhana(MAL-USM); Magidin, Arturo(1-LA); Morse, Robert Fitzgerald(1-EVAN-ECS)
Two generator p-groups of nilpotency class 2 and their conjugacy classes. (English summary) 
Publ. Math. Debrecen 81 (2012), no. 1-2, 145–166. 
A: This is not quite what you were asking, since it involves the lower central series instead of the upper one, but Miech classified the $2$-generator $p$-groups with cyclic commutator subgroup and $p$ odd in:
Miech,  R.J. On $p$-groups with a cyclic commutator subgroup J. Austral. Math. Soc. 20 no. 2 (1975), 178-198, MR0404441 (53 #8243).
Their nilpotency class can be arbitrarily large, though of course they are all metabelian. As far as I know, no similar classification exists for the $p=2$ case. 
A: (Addition to Isaacs' example) Let $G$ be a $p$-group and $d$ the minimal degree of representation of $G$ by permutations. Let $d<=p^n$. where $n$ is as small as possible. Then $G$ is a subgroup
of $\Sigma_{p^n}$, a Sylow $p$-subgroup of the symmetric group of degree $p^n$. Since the center of $\Sigma_{p^n}$ has order $p$, we conclude that $G$ is a subgroup of a $p$-group with center of order $p$.
A: It is easy to prove that if a nonabelian $p$-group $G$ of exponent $>p$ contains $<p$ maximal abelian subgroups of exponent $>p$, then the Hughes subgroup of $G$ is abelian of index $p$.
PROBLEM. Classify the nonabelian $p$-groups of exponent $>p$ containing exactly $p$ maximal abelian subgroups of exponent $>p$.
If $G$ satisfies the problem, then $\exp(G)=p^2$ (Z. Janko). Moreover, all such $G$ are classified for $p=2$.
