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×#26757 |
| 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 | |
664 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)(24,27)(25,28)(26,29)(30,37)(31,35)(32,34)(33,36);(7,10)(11,13)(22,24,25,31)(23,27,28,35)(26,30,33,32)(29,37,36,34);(6,8,9,12)(7,11,10,13)(15,18)(19,21)(23,28)(24,30)(26,33)(27,34)(31,32)(35,37);(7,10)(11,13)(14,16,17,20)(15,19,18,21)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19745);