Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
534 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)(6,7,12,17)(8,13,19,35)(9,22,23,57)(10,26,31,62)(11,32,29,16)(14,38,39,24)(15,41,44,30)(18,46,47,56)(20,51,27,61)(21,54,50,37)(25,43,33,58)(28,40,60,45)(34,59,65,48)(36,64,42,53)(49,67,55,69)(52,66,63,68);(6,8)(7,13)(9,18)(10,27)(11,21)(12,19)(14,34)(15,42)(16,37)(17,35)(20,31)(22,56)(23,47)(24,59)(25,49)(26,61)(28,63)(29,50)(30,64)(32,54)(33,55)(36,44)(38,48)(39,65)(40,66)(41,53)(43,69)(45,68)(46,57)(51,62)(52,60)(58,67);(6,9,11,33,12,23,29,25)(7,14,16,45,17,39,32,40)(8,18,21,55,19,47,50,49)(10,28,30,38,31,60,41,24)(13,34,37,68,35,65,54,66)(15,43,26,57,44,58,62,22)(20,52,53,59,27,63,64,48)(36,67,51,56,42,69,61,46);(8,19)(13,35)(18,47)(20,27)(21,50)(34,65)(36,42)(37,54)(46,56)(48,59)(49,55)(51,61)(52,63)(53,64)(66,68)(67,69)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26646);