By explicit machine computation we obtain the mod– cohomology ring of the third Conway group . It is Cohen–Macaulay, has dimension 4, and is detected on the maximal elementary abelian –subgroups.
"The mod–$2$ cohomology ring of the third Conway group is Cohen–Macaulay." Algebr. Geom. Topol. 11 (2) 719 - 734, 2011. https://doi.org/10.2140/agt.2011.11.719