Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #92586 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C7×C9 ⋊ D4 |
| Sylow | C9 ⋊ D28 |
| Sylow | C7 ⋊ #611 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×D9 |
| commutator, Fitting, cosocle | C2×C7×C9 | C22 |
| Frattini | C2×C3 | D3×D7 |
| socle | C2×C3×C7 | C2×D3 |
| hypocenter | C7×C9 | D4 |
| LCS | G ▷ C2×C7×C9 ▷ C7×C9 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4,6,7,5,3)(9,11)(10,14)(12,13)(15,16)(17,18,19,20);(2,3)(4,5)(6,7)(8,13,15,14,9,11,10,16,12)(18,20)Polycyclic presentation:
2^3,3,7,3|b^2=c,d^3=f^2,b^a=bc,e^a=e^6,d^b=d^2f,f^b=f^257 conjugacy classes — view the full character table.
| maximal subgroup of | #11758981, #11758982, #11758984, … |
| maximal normal subgroup of | #11758981, #11758982, #11758984, … |
| maximal quotient of | #11758863, #11758865, #11758867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 18);