TITLE: Is there a way to prove, that $2$-generated groups are rare among finite groups?
QUESTION [9 upvotes]: Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all } 2 \text{-generated groups of order less than }n}{\text{the number of all groups of order less than } n} = 0$?

This statement is implied by a well known conjecture:

$\lim_{n \to \infty} \frac{\text{the number of all groups of nilpotency class } 2 \text{, exponent } 4 \text{ and order less than }n}{\text{the number of all groups of order less than } n} = 1$

(because the $2$-generated relatively free group of nilpotency class $2$ and exponent $4$ is finite)
However, the problem with aforementioned statement is, that it is...
...well, just a conjecture.
Is there a way to strictly prove the main statement of the question?

REPLY [12 votes]: Lubotzky proved that the number of $2$-generator groups of order at most $n$ (or order exactly $n$) is bounded by $n^{A \log n}$ for some constant $A$. The number of $2$-groups of order $2^m$ with $n \ge 2^m > n/2$ is (corrected, see Will Sawin's comment) $n^{B \log^2 n}$ for some explicit constant $B$. So you win! Using Lubotzky's theorem, you still win even if $2$-generator groups are replaced by $d$-generator groups for any fixed $d$.          
Here is Lubotzky's paper:

Enumerating Boundedly Generated Finite Groups, Journal of Algebra 238 (2001) pp 194–199. doi:10.1006jabr.2000.8650, core.ac.uk version.