Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#12667 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1770 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×M32 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,16,12,17,26,32,10,15,23,31,25,9,14,24)(8,13,19,28,20,29,34,37,21,18,27,33,35,22,30,36);(6,8)(7,13,15,18)(9,22,17,29)(10,21)(11,19)(12,20)(14,30)(16,28,31,33)(23,27)(24,36,32,37)(25,35)(26,34);(6,9,23,32,12,7,14,31,10,17,11,24,25,15,26,16)(8,18,19,33,20,22,34,36,21,13,27,28,35,29,30,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5655);