Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
658 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)(11,16)(13,24)(14,25)(17,26)(18,27)(19,31)(20,30)(22,37)(23,36)(28,29)(32,35)(33,38)(34,41)(39,40);(10,11,17,31,14,19,13,16,15,20,24,29,25,28,26,30)(12,18,32,41,23,34,22,27,21,33,37,40,36,39,35,38);(10,12)(11,18)(13,22)(14,23)(15,21)(16,27)(17,32)(19,34)(20,33)(24,37)(25,36)(26,35)(28,39)(29,40)(30,38)(31,41);(12,21)(18,33)(22,35)(23,36)(27,38)(32,37)(34,39)(40,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26399);