Answer by Derek Holt for Growth rate of orders of perfect groups
The bounds$$n^{l(n)^2/108-cl(n)} \le {\rm Perf}(\le n) \le n^{l(n)^2/48 + l(n)},$$where $c$ is a constant and $l(n)= \log_2(n)$,are proved in this paper: "Enumerating Perfect Groups" by D. F.Holt, J....
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Answer by diracdeltafunk for Growth rate of orders of perfect groups
I'm not sure what is known about this function, but you might be interested to know that it has (basically) been computed for small values of $n$.GAP comes with a library of finite perfect groups,...
View ArticleGrowth rate of orders of perfect groups
Is anything known about the function $P(n)$ where,$$P(n)=|\{ m\leq n :\text{There is a perfect group of order } m\}|.$$Like asymptotics or a good upper bound?
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