Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽7) (order 504) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C8×C7⋊C9 |
| unambiguous semidirect product | (C7×C8)⋊C9 |
| derived, Sylow | C7 ⋊ C8×C9 |
| Hall | C7×C8 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C8×C9 |
| Frattini | C3×C4 | C2×ΓL(1,𝔽8) |
| Fitting | C3×C7×C8 | C3 |
| socle, prime radical | C2×C3×C7 | C3×C4 |
| cosocle | C3×C4×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C8 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,3,4)(5,7,6)(8,9,11,10,12,14,13,15,16)(17,18,19,21,20,22,23,24);(1,2,3,5,4,6,7)120 conjugacy classes — view the full character table.
| maximal subgroup of | (C7⋊C9)⋊C16, C16×C7⋊C9, C8×D7⋊C9, … |
| maximal normal subgroup of | (C7⋊C9)⋊C16, C16×C7⋊C9, C8×D7⋊C9, … |
| maximal quotient of | C16×C7⋊C9, C2×C8×C7⋊C9, C8×C7⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 2);