Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#62635 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25×C5 |
| socle | C2×C5 | C23×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 69 points:
(1,2,3,4,5)(6,7)(8,19)(9,15)(10,16)(11,29)(12,33)(13,18)(14,22)(17,31)(20,44)(21,45)(23,46)(24,39)(25,40)(26,53)(27,56)(28,43)(30,36)(32,58)(34,42)(35,49)(37,59)(38,50)(41,55)(47,65)(48,66)(51,68)(52,63)(54,61)(57,64)(60,67)(62,69);(6,8)(7,14)(9,20)(10,21)(11,30)(12,34)(13,23)(15,35)(16,36)(17,42)(18,38)(19,29)(22,31)(24,47)(25,48)(26,54)(27,57)(28,50)(32,59)(33,37)(39,60)(40,61)(41,64)(43,45)(44,53)(46,58)(49,55)(51,69)(52,67)(56,62)(63,66)(65,68);(6,9)(7,15)(8,20)(10,25)(11,26)(12,27)(13,24)(14,35)(16,40)(17,41)(18,39)(19,44)(21,48)(22,49)(23,47)(28,52)(29,53)(30,54)(31,55)(32,51)(33,56)(34,57)(36,61)(37,62)(38,60)(42,64)(43,63)(45,66)(46,65)(50,67)(58,68)(59,69);(9,24)(15,39)(20,47)(25,27)(26,51)(35,60)(40,56)(41,63)(44,65)(48,57)(49,67)(52,55)(53,68)(54,69)(61,62)(64,66);(6,10,13,12)(7,16,18,33)(8,21,23,34)(9,25,24,27)(11,28,32,31)(14,36,38,37)(15,40,39,56)(17,29,43,58)(19,45,46,42)(20,48,47,57)(22,30,50,59)(26,52,51,55)(35,61,60,62)(41,53,63,68)(44,66,65,64)(49,54,67,69)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55623);