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×#11108 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
866 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)(10,11,15,19,16,20,14,30)(12,21,24,32,25,36,23,17)(13,26,28,18,29,33,27,34)(22,31,38,35,39,41,37,40);(6,7)(8,9)(10,12,14,23,16,25,15,24)(11,17,19,21,20,32,30,36)(13,22,27,37,29,39,28,38)(18,31,33,35,34,41,26,40);(7,9)(10,13)(11,18)(12,22)(14,27)(15,28)(16,29)(17,31)(19,33)(20,34)(21,35)(23,37)(24,38)(25,39)(26,30)(32,41)(36,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4096);