Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18643 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
906 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)(8,16,19,20)(9,14,10,13)(12,17,21,18)(22,23,27,31)(24,32,35,36)(25,28,26,30)(29,34,37,33);(6,8,11,19)(7,12,15,21)(9,18,10,17)(13,16,14,20)(23,28)(25,26)(29,36)(30,31)(32,37)(33,34);(22,24)(23,29)(25,33)(26,34)(27,35)(28,36)(30,32)(31,37);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,20,21,16)(22,25,27,26)(23,30,31,28)(24,33,35,34)(29,32,37,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11631);