Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33810 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23)(24,33)(25,30)(26,29)(27,40)(28,34)(31,39)(32,38)(35,41)(36,43)(37,45)(42,51)(44,49)(46,53)(47,52)(48,50);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24)(23,28)(25,34)(26,35)(27,36)(29,44)(30,33)(31,42)(32,43)(37,49)(38,50)(39,52)(40,48)(41,45)(46,47)(51,53);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(23,29)(24,34)(26,39)(27,38)(28,41)(30,45)(31,32)(33,44)(35,51)(37,53)(40,46)(42,48)(43,47)(49,52);(7,13)(11,14)(12,21)(19,20)(23,30)(24,34)(29,45)(31,46)(32,40)(35,49)(36,50)(51,52)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26798);