Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19010 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,15)(10,14)(11,13)(12,17)(18,21)(19,20)(23,28)(25,26)(29,36)(30,33)(31,34)(35,37);(6,8)(7,12)(9,19)(10,17)(11,18)(13,20)(14,21)(15,16)(22,23)(24,29)(25,28)(26,33)(27,30)(31,35)(32,37)(34,36);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(22,24)(23,29)(25,31)(26,34)(27,32)(28,35)(30,37)(33,36);(7,13)(12,20)(14,15)(16,21)(22,25,27,26)(23,28,30,33)(24,31,32,34)(29,35,37,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11998);