Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19866 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×D8 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,12)(8,10)(13,20)(14,18)(15,19)(16,21)(22,23,27,30)(24,28,26,34)(25,32,33,36)(29,35,37,31);(6,7,11,14)(8,12,10,18)(9,16,17,20)(13,19,21,15)(22,24,27,26)(23,28,30,34)(25,31,33,35)(29,36,37,32);(6,8,11,10)(7,12,14,18)(9,15,17,19)(13,20,21,16)(22,25)(23,29)(24,31)(26,35)(27,33)(28,36)(30,37)(32,34);(6,9,11,17)(7,13,14,21)(8,15,10,19)(12,20,18,16)(22,24,27,26)(23,28,30,34)(25,31,33,35)(29,36,37,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12854);