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×#10902 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
670 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)(10,11,14,20,16,21,15,19)(12,22,25,33,26,32,24,17)(13,27,28,35,30,34,29,18)(23,37,39,41,40,31,38,36);(10,12,15,24,16,26,14,25)(11,17,19,32,21,33,20,22)(13,23,29,38,30,40,28,39)(18,31,34,41,35,37,27,36);(6,8,9,7)(10,13,14,28,16,30,15,29)(11,18,20,27,21,35,19,34)(12,23,25,39,26,40,24,38)(17,31,22,36,33,37,32,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3890);