Efficient power-commutator presentation gives log-size encoding

Statement
Suppose $$p$$ is a prime number and $$P$$ is a group of prime power order with underlying prime $$p$$. In other words, the order of $$P$$ is $$p^n$$ for some positive integer $$n$$. Then, any efficient power-commutator presentation of $$P$$ can be used to give a log-size encoding of $$P$$, i.e., an encoding where the code-words for elements take space that is logarithmic in $$p^n$$ and the time taken for the group operations is expressible as a polynomial in the logarithm of $$p^n$$. We can choose the polynomials to be independent of $$p$$ and $$n$$.