Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24296 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1270 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)(9,24)(10,30)(11,34)(12,17)(13,18)(14,22)(15,28)(16,31)(20,36)(21,49)(23,38)(25,40)(26,41)(27,43)(29,42)(32,44)(33,58)(35,39)(37,51)(45,50)(46,64)(47,61)(48,62)(52,54)(53,68)(55,57)(56,65)(59,63)(60,66)(67,69);(6,8,11,22,13,23,12,35)(7,14,17,19,18,39,34,38)(9,25,27,20,29,47,28,48)(10,21,31,50,33,51,32,59)(15,40,42,62,43,61,24,36)(16,37,30,45,44,49,58,63)(26,53,54,46,56,66,55,67)(41,64,57,68,65,69,52,60);(6,9)(7,15)(8,20)(10,26)(11,27)(12,28)(13,29)(14,36)(16,41)(17,42)(18,43)(19,40)(21,46)(22,47)(23,48)(24,34)(25,35)(30,57)(31,54)(32,55)(33,56)(37,60)(38,61)(39,62)(44,65)(45,64)(49,68)(50,66)(51,67)(52,58)(53,59)(63,69);(6,10,12,32,13,33,11,31)(7,16,34,58,18,44,17,30)(8,21,35,59,23,51,22,50)(9,26,28,55,29,56,27,54)(14,37,38,63,39,49,19,45)(15,41,24,52,43,65,42,57)(20,46,25,53,48,67,47,66)(36,60,61,69,62,68,40,64)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17284);