# Presentation of $H^2(\overline{M}_{0,n},\mathbb{Z})$ as an $S_n$-module?

Let $$\overline{M}_{0,n}$$ be the moduli space of genus zero curves with $$n$$ marked points. Let $$I=\{\{S,S^c\}|S\subset\{1,\dots,n\},|S|\geq2, |S^c|\geq2\}$$ be the set of partitions of $$\{1,\dots n\}$$ into two subsets, each has at least two elements.

Keel shown that the cohomology group $$H^2(\overline{M}_{0,n},\mathbb{Z})$$ is generated by boundary divisor classes $$\delta_{\{S,S^c\}}$$, where $$\{S,S^c\}\in I$$. (Here $$\delta_S$$ is the divisor in $$\overline{M}_{0,n}$$, consisting of curves with two components, marked by $$S$$ and $$S^c$$, and their further degenerations.)

Thus we have a presentation $$\mathrm0\to K\to\bigoplus_{\{S,S^c\}\in I}\mathbb{Z}\cdot\delta_{\{S,S^c\}}\to H^2(\overline{M}_{0,n},\mathbb{Z})\to 0.$$

Keel shown that the kernel $$K$$ is generated by equations $$\sum_{i,j\in S;k,l\notin S}\delta_{\{S,S^c\}}=\sum_{i,k\in S;j,l\notin S}\delta_{\{S,S^c\}}=\sum_{i,l\in S;k,j\notin S}\delta_{\{S,S^c\}},$$ for any four distinct elements $$\{i,j,k,l\}\subset\{1,\dots,n\}$$. These give $$2\cdot{n\choose 4}$$ such equations. But the rank of $$K$$ is only $$\frac{n(n-3)}{2}$$. ($$\#I=2^{n-1}-n-1$$, $$\mathrm{rank}H^2(\overline{M}_{o,n},\mathbb{Z})=2^{n-1}-{n\choose2}-1$$), so these relations are very much dependent..

The question is, would there be a good presentation of $$K$$, my goal is to calculate the group cohomology $$H^1(S_n,H^2(\overline{M}_{0,n},\mathbb{Z}))?$$