${\mathrm{H}}^4({\mathrm{C}\mathrm{o}}_0;\mathbf{Z})=\mathbf{Z}\mathrm{/}24$

We show that the fourth integral cohomology of Conway’s group ${\mathrm{C}\mathrm{o}}_0$ is a cyclic group of order $24$, generated by the first fractional Pontryagin class of the $24$-dimensional representation.