Let $G$ be the subgroup of $\mathrm{GL}\_9(\mathbf{Q})$ 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 $\mathbf{Q}^9$, $G$ is generated by:


* All permutation matrices permuting the basis elements $(a\_1,\dots,a\_8)$ (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.



Identify one group of linear transformations
If I understand your question right, your group $G$ has order $5160960$,
and it has an elementary abelian normal subgroup $N$ of order $2^7$ such
that $G/N \cong {\rm S}\_8$. This can be found with [GAP](https://www.gap-system.org) as follows:



```
gap> A := PermutationMat((1,2),9);;
gap> B := PermutationMat((1,2,3,4,5,6,7,8),9);;
gap> C := [[1,-1,0,0,0,0,0,0,1]/2,
>          [-1,1,0,0,0,0,0,0,1]/2,
>          [1,1,2,0,0,0,0,0,-1]/2,
>          [1,1,0,2,0,0,0,0,-1]/2,
>          [1,1,0,0,2,0,0,0,-1]/2,
>          [1,1,0,0,0,2,0,0,-1]/2,
>          [1,1,0,0,0,0,2,0,-1]/2,
>          [1,1,0,0,0,0,0,2,-1]/2,
>          [1,1,0,0,0,0,0,0,0]];;
gap> G := Group(A,B,C);
<matrix group with 3 generators>
gap> Size(G);
5160960
gap> N := NormalSubgroups(G);
[ Group([  ]), <matrix group of size 2 with 1 generators>, 
  <matrix group of size 128 with 7 generators>, 
  <matrix group of size 2580480 with 9 generators>, 
  <matrix group of size 5160960 with 3 generators> ]
gap> StructureDescription(N[3]);
"C2 x C2 x C2 x C2 x C2 x C2 x C2"
gap> Q := G/N[3];
Group([ (16,24)(17,21)(18,22)(19,23)(20,25)(27,28), (1,6,19,21,12,20,24,13)
(2,16,23,11)(3,5,9,17,22,27,25,14)(4,7,8,10,18,26,28,15), (16,24)(17,21)
(18,22)(19,23)(20,25)(27,28) ])
gap> StructureDescription(Q);
"S8"

```

