Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12990 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C5×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
518 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)(8,9)(10,11,14,19,15,20,29,37,16,21,30,38,28,36,13,18)(12,22,24,34,25,35,23,39,26,40,27,17,31,32,41,33);(10,12,15,31,16,26,28,25)(11,17,20,39,21,34,36,33)(13,27,14,23,29,24,30,41)(18,35,19,22,37,32,38,40);(7,9)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5978);