Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11320 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #397 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
546 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)(7,8)(10,11)(12,20,23,33)(13,17)(14,18)(15,19)(16,22,30,27)(21,36,24,35)(25,31)(26,32)(28,34)(29,39,40,38)(37,41);(6,7)(8,9)(10,12,14,27,15,23,28,22)(11,16,18,33,19,30,34,20)(13,24,25,38,26,21,37,39)(17,29,31,36,32,40,41,35);(6,8,9,7)(10,13,14,25,15,26,28,37)(11,17,18,31,19,32,34,41)(12,21,22,38,23,24,27,39)(16,29,20,35,30,40,33,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4308);