Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#443 ◁ G | |
640 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)(8,20,25,53)(9,26,29,49)(10,30,35,47)(11,18,12,17)(14,38,42,23)(15,36,44,28)(16,33,48,34)(21,54,58,40)(22,59,62,39)(24,63,37,45)(27,65,60,43)(31,50,66,41)(32,46,67,69)(51,57,68,56)(52,55,64,61);(6,8,13,25)(7,14,19,42)(9,21,29,58)(10,31,35,66)(11,24,12,37)(15,39,44,59)(16,45,48,63)(17,41,18,50)(20,47,53,30)(22,60,62,27)(23,33,38,34)(26,64,49,52)(28,56,36,57)(32,55,67,61)(40,65,54,43)(46,51,69,68);(6,9)(7,15)(8,21)(10,27)(11,36)(12,28)(13,29)(14,39)(16,43)(17,49)(18,26)(19,44)(20,51)(22,31)(23,55)(24,57)(25,58)(30,46)(32,34)(33,67)(35,60)(37,56)(38,61)(40,45)(41,52)(42,59)(47,69)(48,65)(50,64)(53,68)(54,63)(62,66);(6,9)(7,15)(8,22)(10,32)(11,28)(12,36)(13,29)(14,40)(16,46)(17,26)(18,49)(19,44)(20,52)(21,31)(23,57)(24,55)(25,62)(27,34)(30,43)(33,60)(35,67)(37,61)(38,56)(39,45)(41,51)(42,54)(47,65)(48,69)(50,68)(53,64)(58,66)(59,63)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26843);