Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #442 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
320 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)(7,8)(10,11)(12,22)(13,25,27,33)(14,28)(15,17)(16,21)(18,23,36,41)(19,30)(20,24)(26,38,29,39)(31,37,40,35)(32,34);(6,7,9,8)(10,12,14,15,16,24,30,32)(11,17,19,22,21,34,28,20)(13,23,29,31,27,41,26,40)(18,33,37,39,36,25,35,38);(6,8,9,7)(10,13,16,27)(11,18,21,36)(12,23,24,41)(14,29,30,26)(15,31,32,40)(17,33,34,25)(19,37,28,35)(20,38,22,39)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6129);