Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17925 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C23×C5 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C5×#431 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,13,10,12,16,19)(8,11,14,18,15,17,20,21)(22,23,27,30,28,24,26,32)(25,29,35,36,33,31,34,37);(22,24,28,23)(25,31,33,29)(26,30,27,32)(34,36,35,37);(6,8,9,14,10,15,16,20)(7,11,13,18,12,17,19,21)(22,25,26,34,28,33,27,35)(23,29,32,37,24,31,30,36);(7,12)(11,17)(13,19)(18,21)(25,33)(29,31)(34,35)(36,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10913);