Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,20)(13,17)(14,26)(15,19)(16,22)(18,29)(21,37)(23,35)(24,33)(25,36)(27,31)(28,34)(30,40)(32,39)(38,41);(10,12,15,23)(11,16,19,31)(13,21,24,38)(14,27,29,22)(17,32,33,30)(18,35,26,20)(25,40,28,39)(34,41,36,37);(7,9)(10,13)(11,17)(12,21)(14,28)(15,24)(16,30)(18,36)(19,33)(20,37)(22,40)(23,38)(25,29)(26,34)(27,39)(31,32)(35,41);(13,24)(17,33)(21,38)(25,28)(30,32)(34,36)(37,41)(39,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16389);