Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C5 · Schur multiplier (H2) = C42 |
| direct product | C5×#12542 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C16 | C42 |
| commutator | C4 | C43×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1726 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C5×C16 ◁ G | |
350 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)(6,7,10,15,11,16,26,37,12,17,27,38,28,39,50,61)(8,13,19,30,21,32,43,54,20,31,42,53,44,55,63,67)(9,14,23,34,25,36,48,59,24,35,47,58,49,60,65,69)(18,29,40,51,22,33,45,56,41,52,62,66,46,57,64,68);(6,8)(7,13)(9,22,24,46)(10,19)(11,21)(12,20)(14,33,35,57)(15,30)(16,32)(17,31)(18,25,41,49)(23,45,47,64)(26,43)(27,42)(28,44)(29,36,52,60)(34,56,58,68)(37,54)(38,53)(39,55)(40,48,62,65)(50,63)(51,59,66,69)(61,67);(6,9)(7,14)(8,18,20,41)(10,23)(11,25)(12,24)(13,29,31,52)(15,34)(16,36)(17,35)(19,40,42,62)(21,22,44,46)(26,48)(27,47)(28,49)(30,51,53,66)(32,33,55,57)(37,59)(38,58)(39,60)(43,45,63,64)(50,65)(54,56,67,68)(61,69)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5530);