Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C23×D3 |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C3 |
| direct product | C3×C8×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C7×C8)⋊C32 |
| unambiguous semidirect product | (C3×C7×C8)⋊C3 |
| Fitting | C3×C7×C8 ⋊ C3 |
| derived, Sylow | C7 ⋊ C32×C8 |
| Hall | C7×C8 ⋊ C32 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3,4)(5,7,6)(8,12,14,9,11,15,10,13);(1,2,3,5,4,6,7)(16,17,18)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 45);