Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10900 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
686 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)(6,7,9,8)(10,11)(12,21)(13,26)(14,30)(15,19)(16,20)(17,23)(18,28)(22,37)(24,33)(25,32)(27,35)(29,34)(31,39)(36,38)(40,41);(10,12,15,23,16,25,14,24)(11,17,19,32,20,33,30,21)(13,22,28,38,29,39,27,40)(18,31,34,41,35,37,26,36);(6,8,9,7)(10,13)(11,18)(12,22)(14,27)(15,28)(16,29)(17,31)(19,34)(20,35)(21,36)(23,38)(24,40)(25,39)(26,30)(32,41)(33,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3888);