Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33617 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#466 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ 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)(8,20)(9,27)(10,16)(11,33)(12,18)(13,19)(14,24)(15,30)(17,21)(22,50)(23,45)(25,46)(26,47)(28,40)(29,60)(31,43)(32,42)(34,41)(35,62)(36,55)(37,48)(38,56)(39,61)(44,53)(49,58)(51,66)(52,57)(54,67)(59,65)(63,68)(64,69);(6,8,21,48,12,25,34,56,13,26,32,55,10,23,11,24)(7,14,33,45,16,36,42,47,19,38,41,46,18,37,17,20)(9,22,49,62,30,53,61,69,31,54,59,68,28,51,29,52)(15,35,58,50,27,57,60,66,40,63,65,67,43,64,39,44);(6,9)(7,15)(8,22)(10,28)(11,29)(12,30)(13,31)(14,35)(16,27)(17,39)(18,43)(19,40)(20,44)(21,49)(23,51)(24,52)(25,53)(26,54)(32,59)(33,58)(34,61)(36,57)(37,64)(38,63)(41,65)(42,60)(45,50)(46,67)(47,66)(48,62)(55,68)(56,69);(6,10,13,12)(7,16,19,18)(8,23,26,25)(9,28,31,30)(11,32,34,21)(14,36,38,37)(15,27,40,43)(17,33,42,41)(20,45,47,46)(22,51,54,53)(24,55,56,48)(29,59,61,49)(35,57,63,64)(39,58,60,65)(44,50,66,67)(52,68,69,62)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26605);