Groups of order $p^3$ are mixed Tate

A natural place to study the Chow ring of the classifying space $BG$, for $G$ a linear algebraic group, is Voevodsky’s triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives $M(BG)$ and $M^c(BG)$, respectively. We show that, for any group $G$ of order $p^3$ over a field of characteristic not $p$ which contains a primitive $p^2$-th root of unity, the motive $M(BG)$ is a mixed Tate motive. We also show that, for a finite group $G$ over a field of characteristic zero, $M(BG)$ is a mixed Tate motive if and only $M^c(BG)$ is a mixed Tate motive.