Symmetric 2-cocycles with values in $\mathbb{C}^\times$

For many finite groups a symmetric $2$-cocycle $\alpha$ ($\alpha(g,h)=\alpha(h,g)$, for all pairs $(h,g)$ of the group) with values in $\mathbb{C}^\times$ is a coboundary. We show using a theoretic arguement and GAP that there is a group of order $64$ having a symmetric $2$-cocycle with a non trivial cohomology class.