Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1610 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)(8,19)(9,25)(10,31)(11,35)(12,17)(13,18)(14,22)(15,29)(16,32)(20,48)(21,37)(23,39)(24,38)(26,46)(27,57)(28,43)(30,42)(33,45)(34,44)(36,50)(40,54)(41,60)(47,51)(49,66)(52,63)(53,62)(55,58)(56,61)(59,65)(64,69)(67,68);(6,8)(7,14)(9,26)(10,21)(11,23)(12,22)(13,24)(15,40)(16,37)(17,38)(18,39)(19,35)(20,28)(25,36)(27,56)(29,50)(30,54)(31,47)(32,53)(33,51)(34,52)(41,64)(42,48)(43,46)(44,62)(45,63)(49,59)(55,61)(57,66)(58,69)(60,68)(65,67);(6,9)(7,15)(8,20)(10,27)(11,28)(12,29)(13,30)(14,36)(16,41)(17,42)(18,43)(19,46)(21,49)(22,26)(23,54)(24,50)(25,35)(31,55)(32,58)(33,59)(34,60)(37,61)(38,40)(39,48)(44,65)(45,57)(47,66)(51,68)(52,69)(53,56)(62,64)(63,67);(6,10,13,34)(7,16,18,45)(8,21,24,52)(9,27,30,60)(11,33,12,32)(14,37,39,63)(15,41,43,57)(17,44,35,31)(19,47,38,62)(20,49,50,69)(22,53,23,51)(25,55,42,65)(26,56,54,68)(28,59,29,58)(36,61,48,67)(40,64,46,66)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16881);