Almost tight asymptotic enumeration of finite $p$-groups: Almost all finite groups are $2$-groups of class $2$

Abstract

Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. We prove, uniformly over all primes $p$, that \[ \log_p f(p^m)=\frac{2}{27}m^3-\frac49m^2+O(m), \] with an absolute implied constant. This improves the classical upper bounds of Sims (\emph{Proc. Lond. Math. Soc.}, 1965) and Newman--Seeley and matches Higman's lower bound (\emph{Proc. Lond. Math. Soc.}, 1960) through the quadratic term. We also prove that \emph{the proportion of $2$-groups of nilpotency class $2$ among all groups of order at most $n$, counted up to isomorphism, tends to $1$ as $n\to\infty$.} These results answer questions~22.3 and~22.19 of Blackburn--Neumann--Venkataraman (\emph{Enumeration of Finite Groups}, 2007). Our proof refines Sims'enumeration framework using graded Lie brackets, constraints on deeper commutator coordinates, and incidence and Fourier estimates for alternating forms. Normalization by changes of generating bases then determines the quadratic term in the upper bound. We also prove that all but a $p^{-\Omega(m^2)}$ fraction of groups of order $p^m$ have a central elementary abelian Frattini subgroup. Combining this concentration with the sharp enumeration bound of Bl\"aser--Li--Qiao--Rogovskyy (\emph{arXiv}, 2025) for $p$-groups of Frattini class $2$ yields the linear error term. The same presentation estimates give fixed-quotient extension bounds which, together with Pyber's prescribed-Sylow theorem and the concentration result, establish domination by $2$-groups of nilpotency class $2$.