I have a discrete group acting on nine numbers $(a_9;a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8)$.

* The entries $a_1$ to $a_8$ can be permuted

* There is additional transformation 
$a_9\to a_1+a_2$;
$a_1\to \frac12(a_1-a_2+a_9)$; $a_2\to \frac12(a_2-a_1+a_9)$;
$a_{i>2}\to a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply what is the order of this group.