Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13468 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #2065 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7,12,38,13,20,37,19)(8,21,53,42,27,50,26,61)(9,28,39,49,34,63,33,46)(10,18,35,16,36,51,11,17)(14,40,43,23,44,25,59,24)(15,32,47,30,48,65,52,31)(22,41,56,69,57,68,62,67)(29,64,60,45,58,66,55,54);(6,8,10,23,12,26,35,40,13,27,36,24,37,53,11,25)(7,14,16,21,19,43,18,42,20,44,17,50,38,59,51,61)(9,29,30,57,33,60,32,62,34,58,31,22,39,55,65,56)(15,45,28,41,47,64,49,67,48,54,63,68,52,66,46,69);(6,9)(7,15)(8,22)(10,32)(11,31)(12,39)(13,34)(14,41)(16,49)(17,46)(18,28)(19,52)(20,48)(21,54)(23,29)(24,58)(25,60)(26,62)(27,57)(30,35)(33,37)(36,65)(38,47)(40,55)(42,64)(43,69)(44,68)(45,50)(51,63)(53,56)(59,67)(61,66)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6456);