Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12911 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #427 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1846 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
266 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)(8,19,23,43)(9,25,28,55)(10,30)(11,17)(12,34)(13,18)(14,21,37,50)(15,27,39,29)(16,32)(20,44)(22,42,24,52)(26,56,58,38)(31,40)(33,60)(35,62)(36,49,61,51)(41,47)(45,66)(46,64)(48,65)(53,68)(54,57,69,59)(63,67);(6,8,10,21,12,24,32,51,13,23,33,50,11,22,31,49)(7,14,16,19,17,36,30,42,18,37,40,43,34,61,60,52)(9,20,26,45,29,47,59,67,28,48,58,68,27,46,57,62)(15,35,38,41,25,53,54,44,39,63,56,64,55,66,69,65);(6,9)(7,15,18,39)(8,20)(10,26)(11,27)(12,29)(13,28)(14,35,37,63)(16,38,40,56)(17,25,34,55)(19,41,43,64)(21,45)(22,46)(23,48)(24,47)(30,54,60,69)(31,57)(32,59)(33,58)(36,53,61,66)(42,44,52,65)(49,62)(50,68)(51,67)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5899);