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×#19001 |
| 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 | |
960 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)(22,23)(24,30)(25,33)(26,28)(27,31)(29,35)(32,37)(34,36);(6,8)(7,12)(9,19)(10,17)(11,18)(13,20)(14,21)(15,16)(22,24,26,32)(23,27,28,36)(25,34,35,31)(29,37,33,30);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(25,35)(29,33)(30,37)(31,34);(7,13)(12,20)(14,15)(16,21)(23,28)(27,36)(29,33)(30,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11989);