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Let denote by $V = V(c, s, t)$ the class of all finite $p$-groups such that $class(G) \leq c$, $expZ(G) \leq p^s$, and $G/Z(G)$ has a maximal normal abelian subgroup of rank $\leq t$.

The class $V$ contains infinitely many $p$-groups since it contains $G \times (Z/(p))^k$, whenever $G$ is contained in $V$.

Is it true that $V$ contains infinitely many p-groups without an abelian diraect factor?

More particularly, is it true that $V$ contains infinitely many p-groups $G$ in which the center is contained in the frattini subgroup?

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Ok, I think that I have an answer. A p-group G in the class V has exponent at most p^s. Also if $Z(G) \leq \Phi (G)$ then d(G) the minimal number of generators of G equals to $d=d(G/Z(G)$, and by a nice result of Thompson (A. Mann for 2-groups), d can not exceed $t(t+1)/2$. It follows from Zelmanov solution of the restricted Burnside problem that V contains only finitely many p-groups with a center contained in the frattini subgroup. Also It seems to me that a variation of the above argument answers the first question negatively. –  Yassine Guerboussa Jun 19 '13 at 16:58
    
in the first line of the above comment, I mean "...exponent at most $p^{sc}$". –  Yassine Guerboussa Jun 19 '13 at 17:02

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