It is proved by Aner Shalev, that for any finite $p$-group of coclass $r$(and sufficiently large order), there is some severe restrictions on lower central series $(\gamma_i(G))$. For instance, $\gamma_i(G)/\gamma_{i+1}(G)$ has order $p$ for all sufficiently large $i$ (there is an explicite lower bound for such $i$'s depending on $p$ and the coclass $r$).

Is there anologue restrictions on the upper central series?

More generally, what can one deduce about the upper central series, from the coclass theory of $p$-groups?

Powerful $p$-groups and uniserial action play a central role in Coclass Theory, and These tools do not say much about the upper central series. Is there a dual theory based for instance on $p$-central $p$-group ($p$-groups in which every element of order $p$ lies in the center)?


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