Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12602 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
490 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)(6,7,9,8)(10,11)(12,20)(13,24)(14,19)(15,17)(16,21)(18,26)(22,28)(23,33)(25,37)(27,30)(29,35)(31,38)(32,34)(36,39)(40,41);(10,12,13,22,14,23,25,36,15,21,26,34,27,35,38,41)(11,16,18,28,19,29,31,39,17,20,24,32,30,33,37,40);(6,8,9,7)(11,17)(16,20)(18,24)(19,30)(28,32)(29,33)(31,37)(39,40)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5590);