ATLAS: Linear group L2(8)

Order = 504 = 23.32.7.
Mult = 1.
Out = 3.

The following information is available for L2(8):

Standard generators

Standard generators of L2(8) are a and b where a has order 2, b has order 3 and ab has order 7.

Standard generators of L2(8):3 are c and d where c has order 2, d has order 3, cd has order 9 and cdcdcddcdcddcdd has order 7. The last condition is equivalent to: cdcdcddcddcdcdd has order 2. Note that these conditions imply that d is conjugate to the field automorphism that squares the matrix entries in the natural representation of L2(8).

Black box algorithms

To find standard generators for L2(8): To find standard generators for L2(8).3:


An outer automorphism of L2(8) of order 3 may be obtained by mapping (a, b) to (a, bababba).

To obtain our standard generators for L2(8):3 we may take c = babb and d to be the above automorphism.
Conversely, we may take a = cd-1cdcd-1cd-1cdcdcd-1cdc and b = cdcdcd-1cd-1cd-1cdcd-1cd. Note also that a' = c and b' = d-1(cd)3d are equivalent under an automorphism to (a, b).


Presentations for L2(8) and L2(8):3 = R(3) in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)7 = (ababab-1ababab-1ab-1)2 = 1 >.

< c, d | c2 = d3 = (cd)9 = [c, d]9 = (cdcdcd-1cd-1cdcd-1)2 = 1 >.


The representations of L2(8) available are The representations of L2(8):3 available are

Maximal subgroups

The maximal subgroups of L2(8) are as follows. The maximal subgroups of L2(8):3 are as follows.

Conjugacy classes

The following are conjugacy class representatives of L2(8). The following are conjugacy class representatives of L2(8):3.
Main ATLAS page Return to main ATLAS page.

Last updated 29th October 1999,
R.A.Wilson, R.A.Parker and J.N.Bray