Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C8 |
| direct product | C5×#8158 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C24×C5 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,11)(9,13)(10,12)(14,18)(15,17)(16,19)(20,21)(22,23,26,30,27,31,38,44)(24,32,35,46,34,28,39,41)(25,29,36,42,37,43,49,52)(33,45,48,53,47,40,50,51);(6,8,9,14,10,15,16,20)(7,11,13,18,12,17,19,21)(22,24)(23,28,31,32)(25,33)(26,35)(27,34)(29,40,43,45)(30,41,44,46)(36,48)(37,47)(38,39)(42,51,52,53)(49,50);(7,12)(11,17)(13,19)(18,21)(22,25,26,36,27,37,38,49)(23,29,30,42,31,43,44,52)(24,33,35,48,34,47,39,50)(28,40,41,51,32,45,46,53)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1146);