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.