Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #92722 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C22×C32×C7 ⋊ C2 |
| Hall | C32×C7 ⋊ D4 |
| Sylow | C32 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ #638 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊D21 |
| commutator, socle, cosocle | C2×C32×C7 | C22 |
| Fitting | C22×C32×C7 | C2 |
| hypercenter | C22 | C3⋊D21 |
| hypocenter | C32×C7 | D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,6,7,5,3)(9,11)(15,16,17);(2,3)(4,5)(6,7)(8,9)(10,11)(13,14)(16,17);(12,13,14)Polycyclic presentation:
2^3,3^2,7|b^a=bc,d^a=d^2,e^a=e^2,f^a=f^6129 conjugacy classes — view the full character table.
| maximal subgroup of | #11759570, #11759573, D7×#638, … |
| maximal normal subgroup of | #11759570, #11759573, D7×#638, … |
| maximal quotient of | #11759311, #11759313, #11759314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 152);