What can one say about a finite pgroup $G$ in which the center of any maximal subgroup is equal to the center of $G$?. CH. J.Cossey and T.Hawkes, (Sets of ppowers as conjugacy class sizes, Proc. Amer. Math. Soc. 128 (2000), 4951.) proved that for any finite set $S$ of powers of $p$, including $1$, there exists a pgroup (of class 2) whose conjugacy class sizes are exactly the members of $S$. It follows from this that there exists a pgroup (of class 2) satisfying the property in the question. I ask particularly if one can find such a pgroup with class greater than 2.
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