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) = C25 |
| direct product | C5×#20776 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#383 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×M16 ◁ G | |
684 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)(9,21,23,28)(10,15)(11,16)(12,20)(13,22,29,35)(14,24)(18,25,31,36)(19,27)(26,33)(30,32,34,37);(6,8,11,19)(7,12)(9,18,23,31)(10,20,24,33)(13,25)(14,17)(15,27)(16,26)(21,34)(22,32,35,37)(28,30)(29,36);(6,9,10,22,11,23,24,35)(7,13,14,28,16,29,15,21)(8,18,20,32,19,31,33,37)(12,25,17,30,26,36,27,34);(8,19)(12,26)(17,27)(18,31)(20,33)(25,36)(30,34)(32,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13764);