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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 $\mathbb{Z}_8$.

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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).

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  • $\begingroup$ PS you can prove that $w_4$ is non-zero based on the fact that the mod 2 cohomology ring is finitely generated over the subring of all Stiefel-Whitney classes. See the introduction to my own paper, P. Guillot, The computation of Stiefel-Whitney classes, Ann. Inst. Fourier vol 59, 2009. $\endgroup$ – Pierre Jun 20 '14 at 7:02

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