Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#12987 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #121 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1869 |
| prime radical | C5×#575 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ 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)(6,7,11,18,12,19,33,48,13,16,34,35,10,17,30,45)(8,14,23,38,22,37,54,64,21,39,31,46,28,43,59,66)(9,24,27,57,26,42,58,47,25,44,32,60,29,15,40,41)(20,49,51,67,52,36,61,63,53,56,55,69,50,62,68,65);(6,8,11,31,12,22,33,59,13,21,34,23,10,28,30,54)(7,14,35,38,19,37,45,64,16,39,18,46,17,43,48,66)(9,20,32,51,26,52,40,61,25,53,27,55,29,50,58,68)(15,36,41,65,24,56,57,67,42,62,47,63,44,49,60,69);(6,9)(7,15)(8,20)(10,29)(11,32)(12,26)(13,25)(14,36)(16,42)(17,44)(18,47)(19,24)(21,53)(22,52)(23,55)(27,34)(28,50)(30,58)(31,51)(33,40)(35,41)(37,56)(38,65)(39,62)(43,49)(45,57)(46,63)(48,60)(54,68)(59,61)(64,67)(66,69)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5975);