Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #157 |
| automorphism | Aut(G) = C22×C3×AΓL(1,𝔽9) (order 1728) · Inn(G) = #157 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C7×#157 |
| Fitting, socle | C2×C32×C7 ⋊ C4 |
| derived, Sylow | C32 ⋊ C2×C4×C7 |
| Hall | C32×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | #157 |
| commutator, hypocenter | C32 | C2×C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C4 |
| prime radical | C2×C7×C3⋊D3 | C2 |
| cosocle | C3⋊D3 | C22×C7 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C7 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 15 points:
(3,4)(5,6,7,8)(9,10,11,12,13,14,15);(1,2)(3,7,5)84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×#157, (C2×D7)⋊#157, Dic7⋊#157, … |
| maximal normal subgroup of | Dic7×#157, (C2×D7)⋊#157, Dic7⋊#157, … |
| maximal quotient of | C7×#3787, C7×#3788, C4×C7×#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 187);