Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33380 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C22×D8 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,21)(13,19)(14,26)(15,30)(16,18)(17,23)(20,28)(22,38)(24,37)(25,35)(27,33)(29,40)(31,36)(32,39)(34,41);(10,12)(11,17)(13,22)(14,27)(15,31)(16,24)(18,32)(19,33)(20,36)(21,26)(23,28)(25,39)(29,41)(30,34)(35,38)(37,40);(11,18)(17,32)(19,30)(20,35)(21,37)(26,40)(33,34)(36,38);(10,13,16,15)(11,19,18,30)(12,22,24,31)(14,25,29,28)(17,33,32,34)(20,26,35,40)(21,38,37,36)(23,27,39,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26368);