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 $p^{f_p(n)}$ which all groups of order $p^n$ embed into. What is the asymptotic growth of $f_p(n)$ when $n$ tends to infinity?

The question asks in a certain sense for how dense $p$-groups can be "packed together" as subgroups of a larger group.

Let's give an example for illustration: By the bound by Francois Brunault, all groups of order $2^{20}$ embed into a group of order $2^{2^{20}-1}$, which is a number with 315653 decimal digits. On the other hand, by Nick Gill's bound, they do not embed into a group of order $2^{66}$, which is a 20-digit number. Can these bounds be refined?

**Added on Feb 21, 2013:** Even if finding precise asymptotics for $f_p(n)$ turns out to be
delicate, isn't it at least possible to decide whether $f_p(n)$ grows polynomially or
exponentially, or whether its growth rate lies somewhere in between?
Or alternatively, are there reasons to believe that this is a difficult problem?

**Added on Dec 4, 2013:** The question whether it is true that $f_p(n)$ grows faster than polynomially but slower than exponentially when $n$ tends to infinity will appear as Problem 18.51 in:

Kourovka Notebook: *Unsolved Problems in Group Theory*. Editors V. D.
Mazurov, E. I. Khukhro. 18th Edition, Novosibirsk 2014.