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×#33613 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
662 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)(10,11,15,20)(12,22,25,38)(13,28,29,18)(14,32,34,39)(16,35,33,21)(17,36,31,26)(19,37,24,23)(27,41,40,30);(10,12,23,28,16,27,34,36,15,25,37,18,33,40,14,26)(11,17,32,30,21,13,24,38,20,31,39,41,35,29,19,22);(7,9)(10,13)(11,18)(12,24)(14,30)(15,29)(16,31)(17,33)(19,25)(20,28)(21,26)(22,37)(23,38)(27,39)(32,40)(34,41)(35,36);(12,25)(13,29)(17,31)(18,28)(22,38)(26,36)(27,40)(30,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26601);