  The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is the subgroup of
the Schur multiplier $H^2(G,\mathbb Q/\mathbb Z)$ consisting of the cohomology
classes which vanishes after restricting to any abelian subgroup of $G$. We
give a proof of Hopf-type formula for $B_0(G)$ and derive an exact sequence for
cohomological version of the Bogomolov multiplier. Using this exact sequence we
provide necessary and sufficient conditions for the corresponding inflation
homomorphism to be an epimorphism and to be zero map. We provide some
conditions for the triviality of $B_0(G)$ for central product of groups $G$ and
show that the Bogomolov multiplier of generalized discrete Heisenberg groups is
trivial. We also give a complete characterization of groups of order $p^6, p>3$
having trivial Bogomolov multiplier.
