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

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 $$p$$.