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×#19797 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
574 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)(7,11)(9,17)(12,20)(13,18)(14,21)(15,19)(22,23,26,30)(24,27,32,36)(25,28,33,29)(31,34,37,35);(6,7,10,13)(8,14,16,20)(9,18,17,11)(12,19,21,15)(22,24,26,32)(23,27,30,36)(25,31,33,37)(28,34,29,35);(23,28)(25,33)(27,34)(29,30)(31,37)(35,36);(6,8,10,16)(7,12,13,21)(9,19,17,15)(11,20,18,14)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12785);