Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
378 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)(6,7,13,19)(8,20,25,55)(9,26,32,62)(10,17,35,52)(11,16,36,51)(12,37,34,18)(14,39,43,22)(15,44,49,29)(21,56,60,40)(23,42,33,53)(24,41,38,50)(27,63,57,68)(28,48,30,66)(31,46,65,47)(45,58,54,67)(59,61,64,69);(6,8)(7,14)(9,27)(10,33)(11,22)(12,38)(13,25)(15,45)(16,50)(17,20)(18,53)(19,43)(21,30)(23,35)(24,34)(26,61)(28,60)(29,59)(31,67)(32,57)(36,39)(37,42)(40,47)(41,51)(44,64)(46,56)(48,63)(49,54)(52,55)(58,65)(62,69)(66,68);(6,9,10,28,34,65,36,44,13,32,35,30,12,31,11,29)(7,15,16,46,37,66,52,62,19,49,51,47,18,48,17,26)(8,21,22,57,38,64,23,58,25,60,39,27,24,59,33,67)(14,40,20,54,53,69,41,68,43,56,55,45,42,61,50,63)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6431);