Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13123 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #121 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1923 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ 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)(6,7,11,17,12,16,32,35,13,18,33,42,10,31,20,46)(8,19,23,44,24,38,53,63,25,45,54,66,22,14,34,37)(9,26,29,57,30,56,58,67,28,15,39,41,27,40,47,60)(21,48,51,68,52,36,61,62,50,43,64,65,49,55,59,69);(6,8,20,34,10,22,33,54,13,25,32,53,12,24,11,23)(7,14,35,44,31,45,17,37,18,38,46,66,16,19,42,63)(9,21,47,59,27,49,39,64,28,50,58,61,30,52,29,51)(15,36,60,65,56,48,41,62,26,55,67,68,40,43,57,69);(6,9,10,27,13,28,12,30)(7,15,16,40,18,26,31,56)(8,21,22,49,25,50,24,52)(11,29,20,47,33,39,32,58)(14,36,19,43,38,55,45,48)(17,41,35,60,42,57,46,67)(23,51,34,59,54,64,53,61)(37,62,44,65,63,69,66,68)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6111);