Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13035 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1879 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
456 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)(6,7,11,18,36,39,35,16,13,20,34,50,12,19,10,17)(8,21,25,14,37,41,23,51,27,54,61,43,26,48,24,42)(9,28,30,47,38,49,29,46,32,44,33,15,31,45,63,52)(22,40,57,65,62,68,56,64,59,67,60,53,58,66,55,69);(6,8,12,37,13,27,36,26)(7,14,39,42,20,43,19,51)(9,22,38,58,32,59,31,62)(10,25,34,23,35,61,11,24)(15,40,46,68,47,67,52,66)(16,48,50,54,17,41,18,21)(28,53,45,69,44,65,49,64)(29,60,33,56,63,57,30,55);(6,9)(7,15)(8,22)(10,33)(11,29)(12,38)(13,32)(14,40)(16,45)(17,49)(18,28)(19,52)(20,47)(21,53)(23,57)(24,60)(25,56)(26,62)(27,59)(30,35)(31,36)(34,63)(37,58)(39,46)(41,64)(42,68)(43,67)(44,50)(48,69)(51,66)(54,65)(55,61)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6023);