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.