Enumeration of groups of prime power order

Enumeration of groups of prime power order refers to the problem of enumerating all finite groups of order $$p^n$$ where $$p$$ is a prime and $$n$$ is a positive integer.

The problem was first studied by Higman and Sims. They introduced the function:

$$f(n,p)$$

which returns the number of groups of order $$p^n$$. The following facts are known about $$f(n,p)$$:


 * $$f(1,p) = 1$$ for all primes $$p$$
 * $$f(2,p) = 2$$ for all primes $$p$$
 * $$f(3,p) = 5$$ for all primes $$p$$
 * $$f(4,2) = 14$$ and $$f(4,p) = 15$$ for any odd prime $$p$$

For higher $$n$$, $$f(n,p)$$ has not yet been found to have a closed expression. Higman proved the following estimate:

$$f(n,p) = p^{An^3}$$

where $$A$$ depends on $$n$$ and $$p$$, and we have:

$$\frac{2}{27} - \varepsilon_n \le A \le \frac{2}{27} + \varepsilon_n$$

with $$\varepsilon_n$$ independent of $$p$$, and approaching zero as $$n \to \infty$$. In other words, we can loosely say:

$$f(n,p) \simeq p^{2n^3/27}$$