Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#10711 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C22 | C43×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C23×C5×C8 ◁ G | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(10,11)(12,20,22,16)(13,17)(14,19)(15,18)(21,32,34,27)(23,33,36,28)(24,30)(25,29)(26,31)(35,40,41,38)(37,39);(6,7,8,9)(10,12)(11,16)(13,21)(14,23)(15,22)(17,27)(18,20)(19,28)(24,35)(25,34)(26,36)(29,32)(30,38)(31,33)(37,41)(39,40);(10,13,14,24,15,25,26,37)(11,17,19,30,18,29,31,39)(12,21,23,35,22,34,36,41)(16,27,28,38,20,32,33,40)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3699);