  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.
