Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13146 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1934 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
312 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)(6,7,11,17,13,18,12,35)(8,19,22,47,24,48,23,57)(9,25,28,60,30,42,29,62)(10,31,32,37,34,63,33,16)(14,36,39,56,40,55,46,21)(15,41,43,26,44,61,59,27)(20,49,52,65,54,67,53,38)(45,66,68,50,58,64,69,51);(6,8,10,21,11,22,32,36,13,24,34,56,12,23,33,55)(7,14,37,48,35,46,31,47,18,40,16,19,17,39,63,57)(9,20,26,50,28,52,61,64,30,54,27,51,29,53,41,66)(15,38,25,58,59,67,62,68,44,65,42,45,43,49,60,69);(6,9,10,26,11,28,32,61,13,30,34,27,12,29,33,41)(7,15,16,42,35,59,63,60,18,44,37,25,17,43,31,62)(8,20,21,50,22,52,36,64,24,54,56,51,23,53,55,66)(14,38,19,45,46,67,57,69,40,65,48,58,39,49,47,68)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6134);