Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C32×D4 (order 288) · Inn(G) = C1 · Out(G) = C22×C32×D4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C4×C7×C9 |
| multiplicative | (ℤ/551)ˣ, (ℤ/635)ˣ, (ℤ/783)ˣ, (ℤ/1102)ˣ, (ℤ/1270)ˣ, (ℤ/1566)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C3 | C22×C3×C7 |
| socle, prime radical | C22×C3×C7 | C2×C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4)(7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22);(5,6)504 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 39);