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?