Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#11098 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
882 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)(13,24,26,17)(14,27)(15,19)(16,22)(18,28)(21,34,36,32)(23,33)(25,31,40,39)(29,35,38,41)(30,37);(6,7)(8,9)(10,12,14,22,15,23,28,37)(11,16,18,20,19,30,27,33)(13,21,25,35,26,36,40,41)(17,29,31,32,24,38,39,34);(7,9)(10,13)(11,17)(12,21)(14,25)(15,26)(16,29)(18,31)(19,24)(20,32)(22,35)(23,36)(27,39)(28,40)(30,38)(33,34)(37,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4086);