Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33583 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
702 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)(7,9)(10,11,16,21)(12,22,25,39)(13,20,15,18)(14,27,30,35)(17,32,34,24)(19,26,37,29)(23,40,31,36)(28,38,41,33);(6,7,8,9);(10,12)(11,17)(13,23)(14,28)(15,31)(16,25)(18,33)(19,36)(20,38)(21,34)(22,27)(24,29)(26,32)(30,41)(35,39)(37,40);(10,13,16,15)(11,18,21,20)(12,23,25,31)(14,26,30,29)(17,33,34,38)(19,35,37,27)(22,36,39,40)(24,28,32,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26571);