18.51 (2014)

Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest number such that there is a group of order $p^{f_p(n)}$ into which every group of order $p^n$ embeds. Is it true that $f_p(n)$ grows faster than polynomially but slower than exponentially when $n$ tends to infinity?