Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13000 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,14,20,28,36,29,38,16,18,13,19,15,21,31,40)(12,22,25,35,32,39,27,17,24,34,23,41,26,37,30,33);(6,7,8,9)(10,12)(11,17)(13,27)(14,30)(15,32)(16,24)(18,33)(19,37)(20,39)(21,41)(22,38)(23,29)(25,31)(26,28)(34,40)(35,36);(11,18)(13,14)(17,33)(21,36)(23,25)(27,30)(29,31)(35,41)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5988);