Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33523 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
1138 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)(10,11,15,20)(12,17,21,31)(13,16,26,30)(14,27,25,19)(18,34,29,24)(22,28,35,39)(23,38,36,33)(32,41,40,37);(11,16)(13,24)(14,25)(17,28)(18,19)(20,30)(22,37)(23,36)(26,34)(27,29)(31,39)(32,33)(35,41)(38,40);(7,9)(10,12)(11,17)(13,22)(14,23)(15,21)(16,28)(18,32)(19,33)(20,31)(24,37)(25,36)(26,35)(27,38)(29,40)(30,39)(34,41);(12,21)(17,31)(22,35)(23,36)(28,39)(32,40)(33,38)(37,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26511);