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) = C25 |
| direct product | C5×#33442 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C22×D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×C8×D4 |
| prime radical | C5×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×M32 ◁ G | |
642 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)(6,7,11,17,12,16,31,41,13,18,32,42,10,28,30,54)(8,19,22,44,21,14,33,35,23,36,48,60,29,37,55,61)(9,15,26,40,27,39,53,63,24,38,49,62,25,50,52,66)(20,43,47,64,46,34,56,58,45,59,65,69,51,57,67,68);(6,8)(7,14)(9,20)(10,29)(11,22)(12,21)(13,23)(15,34)(16,36)(17,35)(18,37)(19,28)(24,45)(25,51)(26,47)(27,46)(30,55)(31,33)(32,48)(38,57)(39,59)(40,58)(41,60)(42,61)(43,50)(44,54)(49,65)(52,67)(53,56)(62,68)(63,69)(64,66);(6,9,11,26,12,27,31,53,13,24,32,49,10,25,30,52)(7,15,17,40,16,39,41,63,18,38,42,62,28,50,54,66)(8,20,22,47,21,46,33,56,23,45,48,65,29,51,55,67)(14,34,35,58,36,59,60,69,37,57,61,68,19,43,44,64);(9,24)(15,38)(20,45)(25,27)(26,49)(34,57)(39,50)(40,62)(43,59)(46,51)(47,65)(52,53)(56,67)(58,68)(63,66)(64,69)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26430);