Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#24997 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×Q8 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,10)(8,12,13,11)(14,15)(16,18)(17,19)(20,21)(22,23,25,28)(24,30,32,27)(26,29,33,36)(31,37,34,35);(7,10)(11,12)(14,16,17,20)(15,18,19,21)(24,31)(26,33)(27,35)(29,36)(30,37)(32,34);(15,19)(18,21)(22,24,25,32)(23,27,28,30)(26,34,33,31)(29,37,36,35);(6,8,9,13)(7,11,10,12)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17985);