Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #397 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
838 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)(10,11,15,19,16,20,14,21)(12,22,26,34,25,33,24,17)(13,27,30,37,31,18,29,36)(23,35,39,32,38,40,28,41);(6,7)(8,9)(10,12,14,24,16,25,15,26)(11,17,19,22,20,34,21,33)(13,28,29,38,31,39,30,23)(18,35,36,32,27,40,37,41);(7,9)(10,13,16,31)(11,18,20,27)(12,23,25,38)(14,30,15,29)(17,32,34,41)(19,37,21,36)(22,35,33,40)(24,39,26,28)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4122);