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The number of isomorphism classes of groups of order $p^n$ grows so fast $\big (p^{{\frac{2}{27}}n^{3}+O(n^{8/3})} \big)$, that a folklore conjecture asserts that, asymptotically, almost every finite group is a 2-group.

In other words, classifying $p$-groups is a tall order; although special families ought to be appealing.

Are there interesting classes of $p$-groups where the pertinent property is a (specific?) tower of $p$-subgroups of every intermediate power $(1,p,p^2,\ldots,p^n)$?

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    $\begingroup$ What do you mean with "pertinent property" here? Also, every $p$-group has a tower of subgroups of the shape you suggest. $\endgroup$
    – Wojowu
    Oct 15 at 18:57
  • $\begingroup$ @Wojowu That is the heart of the question. (BTW I corrected 'cases' to 'classes', which may clarify my meaning) $\endgroup$ Oct 15 at 19:02
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    $\begingroup$ I'm afraid it doesn't, I'm still not sure what it is that you are asking for. $\endgroup$
    – Wojowu
    Oct 15 at 19:23

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