Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3×C4×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C2×C4×C7)⋊C32 |
| unambiguous semidirect product | (C2×C3×C4×C7)⋊C3 |
| Fitting | C2×C3×C4×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C2×C32×C4 |
| Hall | C2×C4×C7 ⋊ C32 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 16 points:
(6,7,8,9)(11,12,13)(14,16,15);(1,2)(3,4,5)(10,11,12,14,13,15,16)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 123);