Take $G$ to be a finite group. The projective representations of $G$ are classified by the group cohomology $H^2(G,U(1))$. We focus on the restriction map

$H^2(G,U(1)) \xrightarrow[]{\rm restriction} \oplus_{B\subseteq G}~ H^2(B,U(1))$,

where $\oplus_{B\subseteq G}$ means the direct sum over all bicyclic subgroups $B = Z_m × Z_n$ of $G$. The kernel of this restriction map is defined as the Bogomolov multiplier $B_0(G)$ of $G$, namely

$B_0(G):={\rm Ker}(~H^2(G,U(1)) \xrightarrow[]{\rm restriction} \oplus_{B\subseteq G}~ H^2(B,U(1))~) $.

My question is what is the simplest finite group $G$ with a non-trivial $B_0(G)$? And how do I express the corresponding non-trivial element of $B_0(G)$ (as a function $G \times G \rightarrow U(1)$)?

Take $G$ to be a finite group. The projective representations of $G$ are classified by the group cohomology $H^2(G,U(1))$. Here $G$ has a trivial action on $U(1)$. We focus on the restriction map

$H^2(G,U(1)) \xrightarrow[]{\rm restriction} \oplus_{B\subseteq G}~ H^2(B,U(1))$,

where $\oplus_{B\subseteq G}$ means the direct sum over all bicyclic subgroups $B = Z_m × Z_n$ of $G$. The kernel of this restriction map is defined as the Bogomolov multiplier $B_0(G)$ of $G$, namely

$B_0(G):={\rm Ker}(~H^2(G,U(1)) \xrightarrow[]{\rm restriction} \oplus_{B\subseteq G}~ H^2(B,U(1))~) $.

My question is what is the simplest finite group $G$ with a non-trivial $B_0(G)$? And how do I express the corresponding non-trivial element of $B_0(G)$ (as a function $G \times G \rightarrow U(1)$)?