  Direct powers of perfect groups admit more concise presentations than one
might naively suppose. If $H_1G=H_2G=0$, then $G^n$ has a presentation with
$O(\log n)$ generators and $O(\log n)^3$ relators. If, in addition, there is an
element $g\in G$ that has infinite order in every non-trivial quotient of $G$,
then $G^n$ has a presentation with $d(G) +1$ generators and $O(\log n)$
relators. The bounds that we obtain on the deficiency of $G^n$ are not monotone
in $n$; this points to potential counterexamples for the Relation Gap Problem.
