Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
890 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)(8,14)(9,17)(10,12)(11,15)(13,19)(16,21)(18,20)(22,23,27,30)(24,31,33,36)(25,29,26,34)(28,35,37,32);(6,8,10,16)(7,11,12,20)(9,18,19,15)(13,21,17,14)(22,24,27,33)(23,28,30,37)(25,32,26,35)(29,36,34,31);(6,7)(8,11)(9,13)(10,12)(14,15)(16,20)(17,19)(18,21)(25,26)(29,34)(31,36)(32,35);(7,12)(11,20)(13,17)(14,21)(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, 11455);