Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #481 |
| 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 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
496 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)(7,9)(10,11,14,20,31,33,28,19,16,18,29,37,15,21,13,27)(12,22,25,17,32,35,23,39,24,40,41,34,26,38,30,36);(6,7,8,9)(10,12,15,32,16,24,31,26)(11,17,33,36,18,34,21,39)(13,25,29,23,28,41,14,30)(19,38,37,40,27,35,20,22);(11,18)(13,28)(14,29)(17,34)(21,33)(23,30)(25,41)(36,39)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5997);