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×#26758 |
| 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 ◁ C4×C5×C4⋊C4 ◁ G | |
672 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)(8,11,12,13)(14,15)(16,19)(17,18)(20,21)(22,23,25,28)(24,30,32,27)(26,29,34,37)(31,36,33,35);(7,10)(11,13)(22,24,25,32)(23,27,28,30)(26,31,34,33)(29,35,37,36);(6,8,9,12)(7,11,10,13)(15,18)(19,21)(24,31)(26,34)(27,35)(29,37)(30,36)(32,33);(7,10)(11,13)(14,16,17,20)(15,19,18,21)(23,28)(27,30)(29,37)(35,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19746);