# Tower of $p$-groups

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 $$(1,p,p^2,\ldots,p^n)$$?

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