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×#24934 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,18)(17,19)(20,21);(7,10)(11,13)(14,16,17,20)(15,18,19,21)(22,23)(24,31)(25,29)(26,34)(27,30)(28,32)(33,36)(35,37);(6,8,9,12)(7,11,10,13)(15,19)(18,21)(22,24)(23,28)(25,32)(26,35)(27,33)(29,36)(30,37)(31,34);(7,10)(11,13)(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17922);