Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#20788 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2283 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C5×#405 ◁ G | |
608 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)(8,17,20,13)(9,14)(10,15)(11,16)(12,19,26,34)(18,32,21,27)(22,29)(23,30)(24,31)(25,33,28,35)(36,37);(6,8)(7,12,16,26)(9,21)(10,19)(11,20)(13,15,17,31)(14,28,30,25)(18,23)(22,35)(24,34)(27,37,32,29)(33,36);(6,8)(7,13)(9,18)(10,19)(11,20)(12,15)(14,27)(16,17)(21,23)(22,33)(24,34)(25,29)(26,31)(28,37)(30,32)(35,36);(6,9,10,22,11,23,24,36)(7,14,15,29,16,30,31,37)(8,18,19,33,20,21,34,35)(12,25,17,32,26,28,13,27)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13776);