Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C4×#120 (order 768) · Inn(G) = C22 · Out(G) = C2×C3×C4×D4 |
| homology | Abelianization (H1) = C2×C4×C5×C7 · Schur multiplier (H2) = C22 |
| direct product | C5×C7×C22⋊C4 |
| unambiguous semidirect product | C22⋊(C4×C5×C7) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5×C7 | C22 |
| commutator | C2 | C2×C4×C5×C7 |
| Frattini, cosocle | C22 | C22×C5×C7 |
| prime radical | C23×C5×C7 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C7 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14,16,18)(15,17,19,20);(14,17)(18,20)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 81);