Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13539 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,12)(8,13,9,11)(14,15)(16,18)(17,19)(20,21)(22,23,25,27)(24,28,31,35)(26,29,33,34)(30,36,37,32);(6,8,10,9)(7,11,12,13)(14,16,17,20)(15,18,19,21)(22,24,25,31)(23,28,27,35)(26,30,33,37)(29,36,34,32);(6,9,10,8)(7,11,12,13)(15,19)(18,21)(23,27)(24,30)(26,34)(28,32)(29,33)(31,37)(35,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6527);