Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21819 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,18)(13,16)(14,17)(15,20)(19,24)(21,23)(22,25)(26,27,29,32,30,33,36,39)(28,31,35,38,34,37,40,41);(10,12,14,20)(11,15,17,18)(13,21,22,19)(16,24,25,23);(7,9)(10,13,14,22)(11,16,17,25)(12,19,20,21)(15,23,18,24)(26,28)(27,31)(29,35)(30,34)(32,38)(33,37)(36,40)(39,41);(11,17)(15,18)(16,25)(23,24)(28,34)(31,37)(35,40)(38,41)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14807);