Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#33789 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,20,25)(16,21,27,42)(17,30,33,64)(18,34,38,63)(19,39,36,24)(22,45,47,31)(23,48,50,37)(26,52,53,75)(28,58,56,74)(29,61,57,44)(32,65,40,49)(35,51,66,46)(41,67,68,54)(43,71,70,60)(55,76,62,72)(59,73,77,69);(6,8,9,13)(7,10,11,12)(14,16)(15,21)(17,26)(18,28)(19,29)(20,27)(22,41)(23,43)(24,44)(25,42)(30,52)(31,54)(32,55)(33,53)(34,58)(35,59)(36,57)(37,60)(38,56)(39,61)(40,62)(45,67)(46,69)(47,68)(48,71)(49,72)(50,70)(51,73)(63,74)(64,75)(65,76)(66,77);(8,13)(10,12)(14,17,20,33)(15,22,25,47)(16,26,27,53)(18,35,38,66)(19,40,36,32)(21,41,42,68)(23,49,50,65)(24,51,39,46)(28,59,56,77)(29,62,57,55)(30,63,64,34)(31,48,45,37)(43,72,70,76)(44,73,61,69)(52,74,75,58)(54,71,67,60);(7,11)(10,12)(16,27)(21,42)(26,53)(28,56)(29,57)(41,68)(43,70)(44,61)(52,75)(54,67)(55,62)(58,74)(59,77)(60,71)(69,73)(72,76)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26777);