A postscript on the quaternion group of order 8.

It's easy to get non-AG presentations as well.

For example,

gap> sl := SpecialLinearGroup( 2 , 3 );;
gap> q := SylowSubgroup( sl , 2 );;

produces Q8 as a group of 2x2 matrices over GF(3):

Subgroup( SL(2,3), [ [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ],
  [ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ],
  [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ] ] )

Then

gap> permq := Operation(q, Elements(q), OnRight);

produces the permutation representation

Group( (1,2)(3,6)(4,8)(5,7), (1,3,2,6)(4,5,8,7), (1,8,2,4)(3,5,6,7) )

with elements

[ (), (1,2)(3,6)(4,8)(5,7), (1,3,2,6)(4,5,8,7), (1,4,2,8)(3,7,6,5),
  (1,5,2,7)(3,4,6,8), (1,6,2,3)(4,7,8,5), (1,7,2,5)(3,8,6,4),
  (1,8,2,4)(3,5,6,7) ]

Nothing fancy here, of course.

C.R.B. Wright