Upper bound on number of groups of prime power order using power-commutator presentations

Statement
Suppose $$p$$ is a prime number and $$n$$ is a natural number. Then, the number of isomorphism classes of groups of order equal to the prime power $$p^n$$ is at most:

$$p^{\binom{n+1}{3}} = p^{\frac{n^3 - n}{6}}$$

Related facts

 * Inductive upper bound on number of groups of prime power order using power-commutator presentations: This is an inductive version of the upper bound that gives a stronger (i.e., smaller) upper bound if we know the number of groups of order $$p^{n-1}$$.
 * Higman-Sims asymptotic formula on number of groups of prime power order tells us that in asymptotic and logarithmic terms, this crude upper bound is not too far from the actual number.