Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10867 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
714 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)(10,11,15,20,16,19,14,29)(12,21,23,32,24,17,30,33)(13,25,27,34,28,40,26,18)(22,36,37,35,38,39,41,31);(6,7,8,9)(10,12)(11,17)(13,22)(14,23)(15,30)(16,24)(18,31)(19,21)(20,32)(25,39)(26,37)(27,41)(28,38)(29,33)(34,35)(36,40);(10,13,14,26,16,28,15,27)(11,18,20,25,19,34,29,40)(12,22,23,37,24,38,30,41)(17,31,32,39,21,35,33,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3855);