Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C24×C5 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
508 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)(10,11,14,19,31,33,29,18,16,21,28,38,15,20,13,27)(12,22,24,17,32,34,23,39,26,40,41,36,25,37,30,35);(10,12,15,32,16,26,31,25)(11,17,33,35,21,36,20,39)(13,24,28,23,29,41,14,30)(18,37,38,40,27,34,19,22);(7,9)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5991);