TITLE: What does "exponent 2 nilpotency class 2" mean?
QUESTION [2 upvotes]: According to the book The Symmetries of Things, p. 208, the number of groups of order 2048 "strictly exceeds 1,774,274,116,992,170, which is the exact number of groups of order 2048 that have 'exponent 2 nilpotency class 2.'" I know what exponent 2 means, and I know what nilpotency class 2 means, but I don't know what is meant by "exponent 2 nilpotency class 2" (clearly not the conjunction of the conditions, as exponent 2 implies elementary abelian), and I haven't been able to find out anywhere. So what does it mean?

REPLY [2 votes]: Let $G$ a finite $p$-group. Define $\mathscr{P}_1(G) = G$, and $\mathscr{P}_{i+1}(G) = [\mathscr{P}_i(G), G]\mathscr{P}_i(G)^p$ for $i > 1$. Then there exists an integer $c$ such that $\mathscr{P}_{c+1}(G) = 1$. If $c$ is the smallest such integer, we say that $G$ has exponent-$p$ class $c$. 
Note that if $G$ has exponent-$p$ class $c$, then $G$ has nilpotency class at most $c$.
See section 2 in this article.
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