Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of Dic126 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C7×C9 ⋊ C2 |
| Hall | C7×C9 ⋊ D4 |
| Sylow | C9 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ #611 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D63 |
| commutator, cosocle | C2×C7×C9 | C22 |
| Frattini | C2×C3 | C2×D21 |
| Fitting | C22×C7×C9 | C2 |
| socle | C2×C3×C7 | C2×D3 |
| hypercenter | C22 | D63 |
| hypocenter | C7×C9 | D4 |
| LCS | G ▷ C2×C7×C9 ▷ C7×C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
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)(8,13,15,14,9,11,10,16,12)(18,20);(2,3)(4,5)(6,7)(9,11)(10,14)(12,13)(15,16)(17,18)(19,20)Polycyclic presentation:
2^3,3,7,3|d^3=f^2,b^a=bc,d^a=d^2f,e^a=e^6,f^a=f^2129 conjugacy classes — view the full character table.
| maximal subgroup of | #11758991, #11758995, D7×#611, … |
| maximal normal subgroup of | #11758991, #11758995, D7×#611, … |
| maximal quotient of | #11758927, #11758929, #11758930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 38);