Let $G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Z})$ defined as follows: Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Z}^9$, $G$ is generated by:

• All permutation matrices permuting the basis $(a_1,\dots,a_8)$ (fixing $a_9$)
• The additional transformation $c$ defined by $a_9\mapsto a_1+a_2$; $a_1\mapsto \frac12(a_1-a_2+a_9)$; $a_2\mapsto \frac12(a_2-a_1+a_9)$; $a_{i>2}\mapsto a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply whether it is finite and if so, to determine its order.