Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18589 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
878 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)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(23,27)(24,30)(25,32)(29,33)(31,34)(35,37);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,23,26,29)(24,28,31,36)(25,33,32,27)(30,35,34,37);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,24,26,31)(23,28,29,36)(25,34,32,30)(27,35,33,37);(8,17)(12,21)(15,20)(16,19)(23,29)(27,33)(28,36)(35,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11577);