Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13126 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1924 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
300 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)(6,7,11,17,12,18,35,50,13,19,36,51,10,16,21,38)(8,20,25,52,23,42,53,48,26,40,33,64,24,14,37,41)(9,27,31,60,30,47,54,49,32,45,34,65,29,15,43,46)(22,55,56,67,28,61,62,69,57,39,59,66,58,44,63,68);(6,8,21,53,10,24,36,25,13,26,35,37,12,23,11,33)(7,14,38,52,16,40,51,41,19,42,50,64,18,20,17,48)(9,28,43,59,29,22,34,62,32,58,54,56,30,57,31,63)(15,44,65,67,45,39,49,68,47,61,60,66,27,55,46,69);(6,9,10,29,13,32,12,30)(7,15,16,45,19,47,18,27)(8,22,23,28,26,57,24,58)(11,34,21,54,36,31,35,43)(14,39,20,44,42,55,40,61)(17,49,38,60,51,46,50,65)(25,59,53,63,33,56,37,62)(41,67,52,69,48,66,64,68)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6114);