Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#18433 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
872 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)(9,16)(11,17)(12,19)(14,20)(22,23,28,25)(24,32,35,37)(26,31,27,30)(29,36,33,34);(6,7)(8,11)(9,12)(10,13)(14,18)(15,17)(16,19)(20,21)(22,24,28,35)(23,29,25,33)(26,34,27,36)(30,32,31,37);(6,8)(7,11)(9,14)(10,15)(12,18)(13,17)(16,20)(19,21)(22,25,28,23)(24,33,35,29)(26,30,27,31)(32,36,37,34);(6,9,10,16)(7,12,13,19)(8,14,15,20)(11,18,17,21)(22,26,28,27)(23,30,25,31)(24,34,35,36)(29,32,33,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11421);