Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#32703 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2782 |
| prime radical | C5×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,15,12,17,10,16)(8,18,21,13,22,29,20,30)(9,23,27,14,25,33,26,32)(19,34,35,31,36,28,24,37);(6,8)(7,13)(9,24)(10,20)(11,21)(12,22)(14,31)(15,29)(16,18)(17,30)(19,27)(23,34)(25,35)(26,36)(28,33)(32,37);(6,9,10,26,12,25,11,27)(7,14,15,33,17,32,16,23)(8,19,21,35,22,36,20,24)(13,28,18,31,30,34,29,37);(8,20,22,21)(9,25)(13,29,30,18)(14,32)(19,35,36,24)(23,33)(26,27)(28,31,34,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25691);