What is the smallest positive integer $n$ such that there are $m$ nonisomorphic groups of order $n$?

Question

This following question given in Gallian's algebra text:

What is the smallest positive integer $n$ such that there are two nonisomorphic groups of order $n$?

The answer for this question given in text book is $n=4$ as $\mathbb Z_4$ and $\mathbb Z_2 \times \mathbb Z_2$ served our purpose.I did this by inspection.

Now i wanted to generalise this question i.e; i wanted to know What is the smallest positive integer $n$ such that there are EXACTLY $m$ nonisomorphic groups of order $n$?

Here inspection does'nt work.So please guide me to get to the result.

Thank you!

Answer 1

Nobody knows. This is an open problem, at this level of generality. In fact, even the weaker question of whether, given a positive integer $m$, there is an $n$ such that the number of groups of order $n$ is equal to $m$ remains open, as far as I know.

There is a very readable survey on the subject that you might enjoy.