Motivic classes of classifying stacks of finite groups and unramified cohomology

Abstract: Combining work of Peyre, Colliot-Th\'el`ene and Voisin, we give the first example of a finite group $G$ such that the motivic class of its classifying stack $BG$ in Ekedahl's Grothendieck ring of stacks over $\mathbb{C}$ is non-trivial and $BG$ has trivial unramified Brauer group.