\}$. More generally, the $i$-th Betti number is given by $$(-1)^i \sum_{Z \in T_i} \sum_{\sigma \in U_Z} (-1)^{d(\sigma)}$$ where $U_Z$ is the set of classes modulo $W$ of the set of simplices of the augmented Folkman complex of the lattice $\mathcal{A}_z$, and $d(\sigma)$ denotes the dimension of a cell. The Folkman complex of a lattice is defined (see [@OT]) as the complex of poset obtained by removing the minimal and maximal elements of the lattice ; when the maximal codimension of the lattice is 1, then the Folkman complex is empty. The augmented Folkman complex is defined by adding to the Folkman complex one $G$-invariant simplex of dimension $-1$. In the case of $G_{31}$ the computation of this formula is doable and we get 0 for the second Betti number.

We then reduce our original complex mod $p^r$, for $p^r$ small enough so that we can encode each matrix entry inside one byte. Then we wrote a C program to compute $H_2(B,\Z_4) = H_2(B,\Z_2) = \Z_2$, $H_2(B,\Z_9) = H_2(B,\Z_3) = \Z_3$ and $H_2(B,\Z_5) = 0$ (the matrix of $d_3$ has size $11065 \times 15300$). Since $G_{31}$ has order $2^{10} .3^2. 5$ and $H_*(P)$ is torsion-free, for $p \not\in \{ 2, 