Does someone know of any works done regarding the density of rank k groups in the collection of groups of order $p^n$ ? i.e. do we know that almost all pgroups are of rank 3 for example?
Thanks in advance for any suggestions
Does someone know of any works done regarding the density of rank k groups in the collection of groups of order $p^n$ ? i.e. do we know that almost all pgroups are of rank 3 for example? Thanks in advance for any suggestions 


In the paper Higman, Graham, Enumerating $p$groups. I. Inequalities. Proc. London Math. Soc. (3) 10 1960 24–30, it is proved that the number of isomorphismclasses of groups of order $p^n$ is $p^{An^3}$, where $2/27−o(n)≤A≤2/15+o(n)$. The upper bound was improved by Sims in Sims, Charles C., Enumerating $p$groups. Proc. London Math. Soc. (3) 15 1965 151–166. where it is shown that $A=2/27+O(n^{−1/3})$. Higman established the lower bound by considering only $p$groups of class 2, in which $G/\Phi(G)$ and $\Phi(G)$ are elementary abelian of orders about $p^{2n/3}$ and $p^{n/3}$ respectively. Although this does not prove that almost all groups of order $p^n$ have rank about $2n/3$, it suggests that that might be the case. 

