Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#8665 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C23×C5 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,13,11,14,9,19)(8,15,18,20,16,21,17,12)(22,23)(24,30)(25,27)(26,34)(28,32)(29,33)(31,37)(35,36);(6,8,9,17,11,16,10,18)(7,12,13,15,14,20,19,21)(23,27)(28,36)(29,34)(30,37);(8,16)(12,20)(15,21)(17,18)(22,24,25,31)(23,28,27,36)(26,35,33,32)(29,37,34,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1653);