Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#27038 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G | |
734 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)(8,12)(9,13)(10,11)(14,15)(16,20)(17,18)(19,21)(22,23,26,29)(24,27,32,36)(25,28,33,37)(30,34,31,35);(7,11)(12,13)(15,18)(19,20)(22,24)(23,27)(25,30)(26,32)(28,34)(29,36)(31,33)(35,37);(6,8,10,9)(7,12,11,13)(14,16,17,21)(15,19,18,20)(24,30)(25,33)(27,34)(28,37)(31,32)(35,36);(6,9,10,8)(7,12,11,13)(15,18)(19,20)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 20026);