Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13021 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1879 |
| prime radical | C5×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
462 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)(6,7,12,19,13,20,36,51,14,21,11,18,35,50,10,17)(8,22,25,44,26,49,24,54,27,52,34,15,39,42,37,43)(9,28,31,55,32,53,29,16,33,46,30,47,40,45,38,48)(23,57,59,64,60,41,65,66,61,67,62,69,63,68,58,56);(6,8,13,39,14,27,35,26)(7,15,20,54,21,44,50,43)(9,23,40,60,33,61,32,63)(10,34,12,24,36,25,11,37)(16,41,55,67,48,68,47,57)(17,49,19,22,51,42,18,52)(28,64,45,66,46,69,53,56)(29,59,31,65,38,62,30,58);(6,9)(7,16)(8,23)(10,31)(11,29)(12,38)(13,40)(14,33)(15,41)(17,46)(18,45)(19,53)(20,55)(21,48)(22,56)(24,62)(25,58)(26,63)(27,61)(28,51)(30,36)(32,35)(34,65)(37,59)(39,60)(42,64)(43,57)(44,68)(47,50)(49,69)(52,66)(54,67)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6009);