Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #92694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C32×C7 ⋊ D4 |
| Sylow | C32 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ #626 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C3⋊D3)⋊D7 |
| commutator, Fitting, socle, cosocle | C2×C32×C7 | C22 |
| hypocenter | C32×C7 | D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
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:
(1,2)(5,6)(8,10)(11,12,14,16,17,15,13);(2,3)(4,5,6)(7,8)(9,10)(12,13)(14,15)(16,17)Polycyclic presentation:
2^3,3^2,7|b^a=bc,d^a=d^2,f^a=f^6,d^b=d^2,e^b=e^251 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#626, D3×#4245, #11759427, … |
| maximal normal subgroup of | D7×#626, D3×#4245, #11759427, … |
| maximal quotient of | #11759217, #11759219, #11759220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 120);