TITLE: Most groups are noncommutative
QUESTION [6 upvotes]: From page 41 of Evan Chen's napkin it states that most groups are noncommutative. This led me to think about an unconventional question: let
$$C_n:=\text{number of nonisomorphic abelian groups of order }n$$
$$G_n:=\text{number of nonisomorphic groups of order }n.$$
Are there any known results on $\limsup_{n\rightarrow\infty}\frac{C_n}{G_n}$ or $\liminf_{n\rightarrow\infty}\frac{C_n}{G_n}$? Notice from here, for $p$ prime we have $C_{p^3}/G_{p^3}=3/5$, so we can deduce
$$\liminf_{n\rightarrow \infty}\frac{C_n}{G_n}\leq\frac{3}{5}$$
and also since $C_{p^2}/G_{p^2}=1$, we have
$$1\leq \limsup_{n\rightarrow \infty}\frac{C_n}{G_n}.$$
My question is, can these bounds be improved, and is determining the exact value possible?
Edit: I realize that since $C_n/G_n\leq 1$ for all $n$, we obviously have $\limsup_{n\rightarrow\infty}C_n/G_n=1$. So forget about that.

REPLY [3 votes]: If $n=2^k$, $k\gg 1$, then $C_n\ll G_n$, so the $\liminf=0$.