Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8817 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #127 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,18)(13,22)(14,16)(15,20)(17,19)(21,24)(23,25)(26,27)(28,34)(29,31)(30,38)(32,36)(33,37)(35,40)(39,41);(10,12,19,25,14,21,13,20)(11,15,22,24,16,23,17,18)(27,31)(32,39)(33,38)(34,40);(11,16)(15,23)(17,22)(18,24)(26,28,30,36,29,35,37,41)(27,32,33,34,31,39,38,40)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1805);