Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×C3×#64241 (order 1728) · Inn(G) = D32 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#624 |
| derived, Sylow | C32 ⋊ C2×C4×C7 |
| Hall | C32×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D32 |
| commutator, hypocenter | C32 | C2×C4×C7 |
| Frattini | C2 | D32×C7 |
| Fitting, socle | C2×C32×C7 | C22 |
| prime radical | C2×C7×C3⋊D3 | C2 |
| cosocle | C2×C32 | C22×C7 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C7 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3)(4,5,6)(7,8,10,9)(11,12,13,14,15,16,17);(1,2,3)(5,6)(7,9,10,8)126 conjugacy classes — view the full character table.
| maximal subgroup of | C7×D32⋊C4, C7×#3773, #11759400, … |
| maximal normal subgroup of | C7×D32⋊C4, C7×#3773, #11759400, … |
| maximal quotient of | C7×#3710, C7×#3712, C7×Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 107);