Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23434 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17)(9,22)(10,27)(11,16)(12,15)(13,19)(14,25)(18,35)(20,29)(21,30)(23,34)(24,32)(26,33)(28,36)(31,37);(6,8)(7,13)(9,23)(10,20)(11,19)(12,21)(14,31)(15,29)(16,30)(17,27)(18,24)(22,28)(25,36)(26,37)(32,34)(33,35);(6,9)(7,14)(8,18)(10,24)(11,25)(12,26)(13,28)(15,32)(16,33)(17,34)(19,23)(20,37)(21,36)(22,27)(29,35)(30,31);(7,15)(13,29)(14,32)(16,27)(17,30)(22,33)(28,35)(31,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16422);