Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19573 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
572 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)(6,7,11,15)(8,16,18,20)(9,19,12,13)(10,21,17,14)(23,27)(24,31)(25,28)(30,35)(32,33)(34,37);(6,8,17,13,11,18,10,19)(7,12,21,20,15,9,14,16)(22,23,28,35,26,30,25,27)(24,29,31,37,32,36,33,34);(6,9)(7,13)(8,14)(10,20)(11,12)(15,19)(16,17)(18,21)(22,24,26,32)(23,29,30,36)(25,33,28,31)(27,34,35,37);(8,18)(9,12)(13,19)(16,20)(24,32)(29,36)(31,33)(34,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12561);