Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#33441 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C16 | C2×D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×C8×D4 |
| prime radical | C5×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C16 ◁ C22×C5×C16 ◁ G | |
646 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)(6,7,11,18,12,17,31,42,13,16,32,41,10,28,30,54)(8,19,22,44,21,14,33,36,23,37,48,61,29,35,55,60)(9,15,25,40,26,39,52,63,27,38,53,62,24,49,51,66)(20,43,46,64,45,34,56,58,47,59,65,69,50,57,67,68);(6,8)(7,14)(9,20)(10,29)(11,22)(12,21)(13,23)(15,34)(16,35)(17,37)(18,36)(19,28)(24,50)(25,46)(26,45)(27,47)(30,55)(31,33)(32,48)(38,57)(39,59)(40,58)(41,60)(42,61)(43,49)(44,54)(51,67)(52,56)(53,65)(62,68)(63,69)(64,66);(6,9,11,25,12,26,31,52,13,27,32,53,10,24,30,51)(7,15,18,40,17,39,42,63,16,38,41,62,28,49,54,66)(8,20,22,46,21,45,33,56,23,47,48,65,29,50,55,67)(14,34,36,58,37,59,61,69,35,57,60,68,19,43,44,64);(7,16)(14,35)(15,38)(17,28)(18,41)(19,37)(34,57)(36,60)(39,49)(40,62)(42,54)(43,59)(44,61)(58,68)(63,66)(64,69)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26429);