Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11126 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #390 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
878 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)(12,22)(13,27,31,18)(14,21)(15,19)(16,20)(17,24)(23,38,40,35)(25,33)(26,34)(28,41,39,32)(29,36,30,37);(6,7,8,9)(10,12,14,24,16,26,15,25)(11,17,19,22,20,33,21,34)(13,28,29,23,31,39,30,40)(18,35,36,32,27,38,37,41);(10,13)(11,18)(12,23)(14,30)(15,29)(16,31)(17,32)(19,37)(20,27)(21,36)(22,35)(24,28)(25,39)(26,40)(33,41)(34,38)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4114);