Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23905 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C5×#3485 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1744 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)(10,11,15,19)(12,16,22,29)(13,17,24,31)(14,25,26,18)(20,32,34,27)(21,35,36,28)(23,30,37,38)(33,40,41,39);(6,7)(8,9)(10,12,15,22)(11,16,19,29)(13,20,24,34)(14,21,26,36)(17,27,31,32)(18,28,25,35)(23,33,37,41)(30,39,38,40);(7,9)(10,13)(11,17)(12,20)(14,23)(15,24)(16,27)(18,30)(19,31)(21,33)(22,34)(25,38)(26,37)(28,39)(29,32)(35,40)(36,41);(10,14)(11,18)(12,21)(13,23)(15,26)(16,28)(17,30)(19,25)(20,33)(22,36)(24,37)(27,39)(29,35)(31,38)(32,40)(34,41)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16893);