Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×C4×D7 |
| Fitting | C32×C4×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C32×C4 |
| Hall | C32×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C4 | D7 |
| commutator, hypocenter | C7 | C2×C32×C4 |
| Frattini | C2 | C2×C32×D7 |
| Fitting | C32×C4×C7 | C2 |
| socle | C2×C32×C7 | C22 |
| prime radical | C2×C32×D7 | C2 |
| cosocle | C2×C7 | C22×C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C32×C4 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 17 points:
(4,5,6)(12,13)(14,15)(16,17);(1,2,3)(7,10,9,8)(11,12,14,16,17,15,13)180 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7×C3⋊C8, C3×#66557, D7×C3⋊(C3⋊C8), … |
| maximal normal subgroup of | C3×D7×C3⋊C8, C3×#66557, D7×C3⋊(C3⋊C8), … |
| maximal quotient of | C32×C8×D7, C32×#1131, C32×C4×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 129);