Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33612 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
694 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)(11,16)(13,17)(14,24)(18,26)(19,27)(20,34)(22,30)(23,36)(25,31)(28,29)(32,41)(33,39)(35,40)(37,38);(10,11,17,29,14,20,25,27,15,19,31,34,24,28,13,16)(12,18,30,38,23,32,35,39,21,33,40,41,36,37,22,26);(7,9)(10,12)(11,18)(13,22)(14,23)(15,21)(16,26)(17,30)(19,33)(20,32)(24,36)(25,35)(27,39)(28,37)(29,38)(31,40)(34,41);(11,19)(12,21)(16,27)(20,28)(22,35)(23,36)(29,34)(30,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26600);