Consider the space $X=BSL(8,\mathbb{C})/(\mathbb{Z}/2)$. The topological Brauer group of $X$ is given by $Br_{top}(X)=Tor(H^{3}(X;\mathbb{Z}))=\mathbb{Z}/2$. I'm studying concepts of topological period and index of a class in $Br_{top}(X)$ and I would like to calculate $ind_{top}(\alpha)$ where $\alpha$ is the nontrivial class in the topological Brauer group of the space given above.

My question is: What should I know about this space that can help me to calculate the index?.

It's known that $per_{top}(\alpha)|ind_{top}(\alpha)$ and that they have the same prime divisors. Since $per_{top}(\alpha)=2$, then $ind_{top}(\alpha)=2^{m}$ for some positive integer $m>0$.