Higman-Sims asymptotic formula on number of groups of prime power order

Statement

Quick version

Let $p$ be a (fixed) prime number. Define $f(n,p)$ as the number of groups of order $p^{n}$ (see also group of prime power order, number of groups of given order). Then:

$f(n,p)=p^{(2/27+o(1))n^{3}}$

Full version

Let $p$ be a (fixed) prime number. Define $f(n,p)$ as the number of groups of order $p^{n}$ (see also group of prime power order, number of groups of given order). Define:

$A(n,p):={\frac {\log f(n,p)}{n^{3}\log p}}$

Then:

$A(n,p)={\frac {2}{27}}+O(n^{-1/3})$

Note that the big-O notation is with respect to $n$ (sending $n\to \infty$ and not with respect to $p$). However, the constants under the big-O notation may depend on $p$.