  The Erd\H{o}s Pentagon problem asks to find an $n$-vertex triangle-free graph
that is maximizing the number of $5$-cycles. The problem was solved using flag
algebras by Grzesik and independently by Hatami, Hladk\'{y}, Kr\'{a}l', Norin,
and Razborov. Recently, Palmer suggested the general problem of maximizing the
number of $5$-cycles in $K_{k+1}$-free graphs. Using flag algebras, we show
that every $K_{k+1}$-free graph of order $n$ contains at most
\[\frac{1}{10k^4}(k^4 - 5k^3 + 10k^2 - 10k + 4)n^5 + o(n^5)\] copies of $C_5$
for any $k \geq 3$, with the Tur\'an graph begin the extremal graph for large
enough $n$.
