The symmetric group $S\_n$ has an $n$-dimensional defining representation, which splits as $n = (n-1) + 1$. Although this representation exists integrally, I would like to think of this as a real representation $S\_n \to O(n)$, or equivalently as an $n$-dimensional real vector bundle on the classifying space $BS\_n$.


Real vector bundles $V \to X$ have Pontryagin classes. The first Pontryagin class $p\_1(V)$ lives in $H^4(X;\mathbb Z)$. When $n \gg 0$, $H^4(BS\_n;\mathbb Z) \cong \mathbb Z/2 \times \mathbb Z/2 \times \mathbb Z/12$. $p\_1$ is additive for direct sum of real vector bundles, and vanishes on trivial bundles.



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> What is the value of the first Pontryagin class of the defining rank-$n$ bundle on $BS\_n$?
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I am happy with answers just in the stable range, but of course feel free to share the unstable answers as well. (I say stable, but of course I expect $p\_1$ depends on $n$, just not $H^4$.)



What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?
If I am not confused, $p\_1(V)=-c\_2(V\otimes {\mathbb C})$. According to Theorem 7.1 in the book *Characteristic Classes and the Cohomology of Finite Groups* by Charles Thomas, $c\_2$ of the standard representation of $S\_n$ has order $12$, for $n$ large enough.


