Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33596 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
674 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)(6,7,13,19)(8,20,22,39)(9,26,29,59)(10,18,12,16)(11,31,34,43)(14,25,37,23)(15,40,42,28)(17,30,45,33)(21,49,50,69)(24,38,53,65)(27,60,35,44)(32,46,62,41)(36,63,64,55)(47,67,66,52)(48,54,61,56)(51,58,57,68);(6,8,10,23,13,22,12,25)(7,14,16,20,19,37,18,39)(9,21,27,51,29,50,35,57)(11,24,30,54,34,53,33,56)(15,36,41,47,42,64,46,66)(17,38,43,48,45,65,31,61)(26,58,44,49,59,68,60,69)(28,52,32,55,40,67,62,63);(6,9)(7,15)(8,21)(10,27)(11,32)(12,35)(13,29)(14,36)(16,41)(17,44)(18,46)(19,42)(20,47)(22,50)(23,51)(24,55)(25,57)(26,31)(28,33)(30,40)(34,62)(37,64)(38,49)(39,66)(43,59)(45,60)(48,68)(52,56)(53,63)(54,67)(58,61)(65,69);(8,22)(14,37)(20,39)(21,50)(23,25)(24,53)(36,64)(38,65)(47,66)(48,61)(49,69)(51,57)(52,67)(54,56)(55,63)(58,68)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26584);