Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33542 |
| 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 | |
1084 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)(8,20,24,48)(9,25,29,40)(10,18,12,16)(11,31,34,43)(14,36,39,23)(15,26,42,28)(17,30,45,33)(21,49,53,67)(22,54,35,44)(27,58,61,41)(32,46,62,38)(37,59,65,69)(47,50,68,52)(51,55,63,64)(56,57,66,60);(6,8)(7,14)(9,21)(10,22)(11,32)(12,35)(13,24)(15,37)(16,38)(17,44)(18,46)(19,39)(20,31)(23,33)(25,55)(26,50)(27,59)(28,52)(29,53)(30,36)(34,62)(40,64)(41,67)(42,65)(43,48)(45,54)(47,56)(49,58)(51,60)(57,63)(61,69)(66,68);(6,9)(7,15)(8,21)(10,26)(11,27)(12,28)(13,29)(14,37)(16,40)(17,41)(18,25)(19,42)(20,47)(22,50)(23,51)(24,53)(30,57)(31,56)(32,59)(33,60)(34,61)(35,52)(36,63)(38,64)(39,65)(43,66)(44,67)(45,58)(46,55)(48,68)(49,54)(62,69);(6,10,13,12)(7,16,19,18)(8,22,24,35)(9,26,29,28)(11,30,34,33)(14,38,39,46)(15,40,42,25)(17,43,45,31)(20,44,48,54)(21,50,53,52)(23,32,36,62)(27,57,61,60)(37,64,65,55)(41,66,58,56)(47,67,68,49)(51,59,63,69)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26530);