TITLE: What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?
QUESTION [14 upvotes]: 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.

What is the value of the first Pontryagin class of the defining rank-$n$ bundle on $BS_n$?

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 $H^4$.)

REPLY [11 votes]: 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.