Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7878 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C23×C5 | C2×C4⋊C4 |
| prime radical | C24×C5 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,20)(13,17)(14,18)(15,19)(16,21)(22,25)(23,24)(26,27,29,32,30,33,36,39)(28,31,35,38,34,37,40,41);(7,9)(10,12,14,22,15,23,13,21)(11,16,17,24,19,25,18,20)(26,28)(27,31)(29,35)(30,34)(32,38)(33,37)(36,40)(39,41);(10,13,15,14)(11,17,19,18)(12,21,23,22)(16,24,25,20)(28,34)(31,37)(35,40)(38,41)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 866);