Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C5×#33340 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C22×D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C22×C22⋊C8 |
| prime radical | C5×#585 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×#131 ◁ G | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,4,5)(6,7,11,17,12,18,32,45,13,19,33,46,10,16,30,43)(8,20,23,48,22,37,52,44,24,42,31,58,29,14,34,36)(9,15,27,40,28,41,57,66,25,38,53,62,26,39,55,64)(21,47,51,67,50,60,68,65,49,63,56,69,54,35,59,61);(6,8)(7,14)(9,21)(10,29)(11,31)(12,22)(13,24)(15,35)(16,42)(17,44)(18,20)(19,37)(23,33)(25,49)(26,54)(27,56)(28,50)(30,52)(32,34)(36,46)(38,60)(39,63)(40,65)(41,47)(43,48)(45,58)(51,53)(55,68)(57,59)(61,62)(64,67)(66,69);(6,9)(7,15)(8,21)(10,26)(11,27)(12,28)(13,25)(14,35)(16,39)(17,40)(18,41)(19,38)(20,47)(22,50)(23,51)(24,49)(29,54)(30,55)(31,56)(32,57)(33,53)(34,59)(36,61)(37,60)(42,63)(43,64)(44,65)(45,66)(46,62)(48,67)(52,68)(58,69);(9,25)(15,38)(21,49)(26,28)(27,53)(35,60)(39,41)(40,62)(47,63)(50,54)(51,56)(55,57)(59,68)(61,65)(64,66)(67,69)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26328);