Why are there lots of groups with order $2^n$?

Question

I've noticed that, in the OEIS sequence A000001, lots of record high values are held by powers of $2.$ The records are held by only $1, 4, 8, 16, 24, 32, 48, 64, 128, 256, 512,$ and $1024.$ The only record holders that aren't powers of two are $24$ and $48.$ Can anyone tell me why this is so?

Answer 1

As already mentioned in the comments, there is a nice post by Aron Amit on Quora to answer this question. Below I will write roughly the same thing said in a slightly different way.

The number of groups of a given order $n$ generally depends on how complicated the prime factorisation of $n$ is. In particular, if $n$ is prime, it is a simple exercise in basic group theory to show that there is only one group of order $n$ up to isomorphism. If $n$ is semiprime, i.e. $n=pq$ for primes $p,q$, then there are at most two groups of order $n$. However, if $n$ has a large number of factors, then the number of groups of the given order explodes.

One can understand this by appreciating how to construct groups of a certain order. Of course, given any two groups $G,H$ of order $m,n$ respectively we can construct the direct product $G\times H$ which has order $mn$. But we can also construct a semidirect product, $G\ltimes_\varphi H$ where $\varphi:H\to\operatorname{Aut}(G)$ is any homomorphism, which also has order $mn$. This is like taking the product of the two groups, but with an added "twist" in the group operation that causes additional interactions between $G$ and $H$. Since we have many choices of $\varphi$, this gives us extra freedom to build groups of order $mn$ starting from $G$ and $H$.

Therefore, if you imagine a group of order $n$ with large number of factors, then intuitively you would expect that there are many ways to choose "factor" groups in such a way that they can be pieced together to a group of order $n$. And there will be many ways to piece them together, due to the availability of choice of $\varphi$ in the semidirect product construction. Inductively, you expect the number of groups to therefore increase rapidly with the number of factors of $n$.

For more, you may want to read about the extension problem in group theory.