It is extremely well-known that $H^\*(BQ\_8;\mathbb{Z})=\mathbb{Z}[\alpha,\beta,\gamma]$ with relations $2\alpha=2\beta=8\gamma=\alpha^2=\beta^2=\alpha\beta-4\gamma=0$, $|\alpha|=|\beta|=2$ and $|\gamma|=4$. I was wondering if anyone can given an explicit reference showing that $c\_2(\sigma)$ generates $H^4(BQ\_8)\cong \mathbb{Z}\_8$, where $\sigma: Q\_8 \hookrightarrow SU(2)$ is the standard representation (i.e., $c\_2(\sigma)=c\_2(E\_\sigma)$, $E\_\sigma= S^\infty\times\_{Q\_8}\mathbb{C}^2)$. 


Any help would be greatly appreciated, as I would like to avoid the following reasoning if possible:


$E\_\sigma$ is the pullback of the canonical quaternionic line bundle $E=S^\infty\times\_{SU(2)}\mathbb{C}^2\rightarrow \mathbb{H}P^\infty$ by the projection $S^3/Q\_8\hookrightarrow BQ\_8 \stackrel{\pi}{\rightarrow}\mathbb{H}P^\infty$, and Leray-Hirsch gives $H^4(\mathbb{H}P^\infty;\mathbb{Z}\_2)\cong H^4(BQ\_8;\mathbb{Z}\_2)\cong\mathbb{Z}\_2$. Looking at $H^4(\mathbb{H}P^\infty;\mathbb{Z})\cong \mathbb{Z}\rightarrow H^4(BQ\_8;\mathbb{Z})\cong \mathbb{Z}\_8$ then shows that $c\_2(\rho)=\pi^\*(c\_2(E))$ must therefore represent an odd number in $\mathbb{Z}\_8$. 



Explicit reference on generator of $H^4(BQ_8;\mathbb{Z})\cong \mathbb{Z}_8$ identified with second Chern class of standard representation
It is enough to show that the reduction mod 2 of that class is not zero. However, reducing $c\_2$ mod 2 gives the Stiefel-Whitney class $w\_4$ of the underlying real representation. That this class is non-zero is classical, and explicitly stated and proved for example in Quillen's "The Mod 2 Cohomology Rings of Extra-special 2-groups and the Spinor Groups", Math. Ann. 194, 197--212 (1971). (A somewhat overkill reference, proving much more).


