Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33837 |
| 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 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(11,17)(12,16)(13,21)(14,26)(18,34)(19,36)(20,33)(22,29)(23,38)(24,37)(25,31)(27,30)(32,39)(35,40);(10,11,15,20)(12,18,16,30)(13,23,25,36)(14,27,29,34)(17,22,33,26)(19,37,38,24)(21,40,31,32)(28,35,41,39);(7,9)(10,12,15,16)(11,18,20,30)(13,21,25,31)(14,22,29,26)(17,27,33,34)(19,35,38,39)(23,40,36,32)(24,28,37,41);(10,13,15,25)(11,19,20,38)(12,21,16,31)(14,28,29,41)(17,32,33,40)(18,35,30,39)(22,37,26,24)(23,34,36,27)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26825);