$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie
group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ preserving a certain quartic form
(see, e.g., here).

Inside this is a discrete subgroup called $\Es(\Z)$, which is the intersection of $\Es$ with
$\mathrm{Sp}_{56}(\Z)$. This group appears in theoretical physics, where it is called the U-duality group and is
the symmetry group of a supergravity theory.

**What is known about the group cohomology of $\Es(\Z)$?** I am interested in knowing the ring structure of
$H^*(\Es(\Z); k)$ where $k = \mathbb Q$ or $\mathbb F_p$, though I only need it up to about degree 6 or 7.
For $\mathbb F_p$ coefficients, if the Steenrod action is known that would also be nice to know.

I don't know what's known about the cohomology of infinite discrete groups; as far as I know, this could be a
straightforward calculation given $H^*(B\Es;\Z)$ (which is known), or it could be totally out of reach right now.
**I would also welcome an answer with that information, and/or where to read more.**

This post imported from StackExchange MathOverflow at 2022-01-03 16:02 (UTC), posted by SE-user Arun Debray