Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
1086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,16,21)(12,22,26,50)(13,18,27,41)(14,30,34,48)(15,35,32,20)(17,37,40,24)(19,44,46,33)(23,49,51,71)(25,54,36,45)(28,57,55,70)(29,60,56,43)(31,47,62,39)(38,63,64,52)(42,67,66,59)(53,72,61,68)(58,69,73,65);(10,12,16,26)(11,17,21,40)(13,23,27,51)(14,31,34,62)(15,36,32,25)(18,38,41,64)(19,45,46,54)(20,47,35,39)(22,48,50,30)(24,44,37,33)(28,58,55,73)(29,61,56,53)(42,68,66,72)(43,69,60,65)(49,70,71,57)(52,67,63,59);(7,9)(10,13)(11,18)(12,23)(14,28)(15,29)(16,27)(17,38)(19,42)(20,43)(21,41)(22,49)(24,52)(25,53)(26,51)(30,57)(31,58)(32,56)(33,59)(34,55)(35,60)(36,61)(37,63)(39,65)(40,64)(44,67)(45,68)(46,66)(47,69)(48,70)(50,71)(54,72)(62,73);(13,27)(18,41)(23,51)(28,55)(29,56)(38,64)(42,66)(43,60)(49,71)(52,63)(53,61)(57,70)(58,73)(59,67)(65,69)(68,72)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26513);