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We give formulas for calculating the unramified Brauer group of a homogeneousspace $X$ of a semisimple simply connected group $G$ with finite geometricstabilizer $\bar F$ over a wide family of fields of characteristic 0. When $k$is a number field, we use these formulas in order to study the Brauer-Maninobstruction to the Hasse principle and weak approximation. We prove inparticular that the Brauer-Manin pairing is constant on $X(k_v)$ for every $v$outside from an explicit finite set of non archimedean places of $k$.

The unramified Brauer group of homogeneous spaces with finite stabilizer