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Let G be a finite group. A natural place to study the Chow ring of the classifying space BG is Voevodsky's triangulated category of motives, inside which Morel-Voevodsky and Totaro have defined motives M.BG/ and Mc.BG/, respectively. We show that, for any group G of order p³ over a field of characteristic not equal to p which contains a primitive p³-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 if Mc.BG/ is a mixed Tate motive. [ABSTRACT FROM AUTHOR]