Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1879 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,17,19,22,11,16,14,18,30,31,13,8,10,21)(9,23,26,15,32,33,24,20,28,36,29,35,27,37,25,34);(6,8,19,18,14,22,13,7)(9,20,27,35,28,34,32,15)(10,21,12,31,11,16,30,17)(23,26,33,25,36,29,37,24);(6,9)(7,15)(8,20)(10,29)(11,26)(12,24)(13,32)(14,28)(16,33)(17,36)(18,35)(19,27)(21,37)(22,34)(23,31)(25,30)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6024);