Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23723 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#399 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
972 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)(6,7,10,14,11,15,17,25)(8,16,19,27,20,31,24,12)(9,21,23,35,22,13,28,30)(18,32,34,29,33,37,36,26);(6,8,17,19,11,20,10,24)(7,12,25,16,15,27,14,31)(9,18,28,34,22,33,23,36)(13,29,35,37,21,26,30,32);(6,9)(7,13)(8,18)(10,23)(11,22)(12,26)(14,30)(15,21)(16,32)(17,28)(19,34)(20,33)(24,36)(25,35)(27,29)(31,37);(9,22)(13,21)(18,33)(23,28)(26,29)(30,35)(32,37)(34,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16711);