Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13171 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(11,17)(12,21)(13,15)(14,22)(18,31)(19,34)(20,26)(23,36)(24,35)(25,37)(27,28)(29,32)(33,41)(38,40);(6,8,9,7)(10,11)(12,18)(13,26)(14,19)(15,17)(16,20)(21,32)(22,30)(23,38)(24,29)(25,33)(27,39)(28,34)(31,36)(35,41)(37,40);(10,12,22,36,15,24,28,37,16,25,27,35,13,23,14,21)(11,18,30,31,17,29,34,40,20,33,39,41,26,38,19,32)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6159);