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) = C24 |
| direct product | C5×#13073 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
478 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)(6,7)(8,9)(10,11,15,20,17,13,16,28)(12,21,25,38,26,37,33,39)(14,29,31,27,22,40,32,19)(18,34,35,24,23,41,36,30);(10,12,22,24,16,25,31,41,17,26,14,30,15,33,32,34)(11,18,19,37,20,36,29,38,13,23,27,21,28,35,40,39);(7,9)(10,13)(11,17)(12,23)(14,19)(15,20)(16,28)(18,26)(21,24)(22,27)(25,35)(29,32)(30,37)(31,40)(33,36)(34,38)(39,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6061);