Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19061 |
| 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 | |
942 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)(8,16)(9,13)(10,14)(11,15)(12,18)(17,20)(19,21)(22,23)(24,31)(25,29)(26,34)(27,30)(28,32)(33,36)(35,37);(6,8)(7,12)(9,17)(10,18)(11,19)(13,16)(14,21)(15,20)(22,24,27,33)(23,28,30,37)(25,32,26,35)(29,36,34,31);(6,8)(7,12)(9,18)(10,17)(11,19)(13,21)(14,16)(15,20)(25,26)(29,34)(31,36)(32,35);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,16,20,21)(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12049);