Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
1090 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)(8,18,21,47)(9,14,23,38)(10,26,31,61)(11,32,29,16)(13,34,36,20)(15,41,44,30)(19,48,49,46)(22,53,33,42)(24,55,28,60)(25,58,54,40)(27,45,63,37)(35,52,65,50)(39,64,43,57)(51,69,59,67)(56,68,62,66);(6,8)(7,13)(9,19)(10,27)(11,33)(12,21)(14,35)(15,42)(16,45)(17,36)(18,26)(20,30)(22,29)(23,49)(24,56)(25,59)(28,62)(31,63)(32,37)(34,41)(38,65)(39,67)(40,68)(43,69)(44,53)(46,60)(47,61)(48,55)(50,57)(51,54)(52,64)(58,66);(6,9)(7,14)(8,19)(10,28)(11,25)(12,23)(13,35)(15,43)(16,40)(17,38)(18,46)(20,52)(21,49)(22,51)(24,31)(26,60)(27,62)(29,54)(30,64)(32,58)(33,59)(34,50)(36,65)(37,66)(39,44)(41,57)(42,69)(45,68)(47,48)(53,67)(55,61)(56,63);(9,23)(14,38)(19,49)(24,28)(25,54)(35,65)(39,43)(40,58)(46,48)(50,52)(51,59)(55,60)(56,62)(57,64)(66,68)(67,69)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26520);