Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
636 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)(8,20,23,46)(9,27,30,44)(10,31,33,42)(11,18,12,17)(14,35,37,24)(15,34,41,29)(16,32,43,21)(22,51,52,68)(25,48,26,47)(28,59,60,40)(36,63,64,69)(38,55,39,49)(45,56,67,54)(50,58,61,66)(53,57,62,65);(6,8,21,49,12,26,33,35,13,23,32,55,11,25,10,24)(7,14,31,48,18,39,43,46,19,37,42,47,17,38,16,20)(9,22,50,69,29,54,60,62,30,52,61,63,34,56,28,53)(15,36,58,51,27,57,59,67,41,64,66,68,44,65,40,45);(6,9)(7,15)(8,22)(10,28)(11,34)(12,29)(13,30)(14,36)(16,40)(17,44)(18,27)(19,41)(20,45)(21,50)(23,52)(24,53)(25,56)(26,54)(31,58)(32,61)(33,60)(35,62)(37,64)(38,65)(39,57)(42,66)(43,59)(46,67)(47,68)(48,51)(49,69)(55,63);(8,23)(14,37)(20,46)(22,52)(24,35)(25,26)(36,64)(38,39)(45,67)(47,48)(49,55)(51,68)(53,62)(54,56)(57,65)(63,69)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26607);