Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C23×C3×AGL(1,ℤ/9) (order 1296) · Inn(G) = D9 · Out(G) = C23×C32 |
| homology | Abelianization (H1) = C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C7×C9⋊C8 |
| unambiguous semidirect product | C9⋊(C7×C8) |
| derived, Sylow | C9 ⋊ C7×C8 |
| Hall | C7×C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | D9 |
| commutator, hypocenter | C9 | C7×C8 |
| Frattini | C3×C4 | D3×C7 |
| Fitting | C4×C7×C9 | C2 |
| socle, prime radical | C2×C3×C7 | Dic3 |
| cosocle | C4×C9 | C2×C7 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C4×C7 | |
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:
(2,4)(3,7)(5,6)(8,9)(10,17,16,15,13,14,12,11)(18,19,20,21,22,23,24);(1,9,4,7,6,5,3,2,8)168 conjugacy classes — view the full character table.
| maximal subgroup of | D63⋊C8, #11758855, D7×C9⋊C8, … |
| maximal normal subgroup of | D63⋊C8, #11758855, D7×C9⋊C8, … |
| maximal quotient of | C7×C9⋊C16, C2×C7×C9⋊C8, C7×C27⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 3);