Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13270 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1985 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
316 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)(6,7,11,17,12,16,28,31,13,18,29,35,10,24,20,36)(8,19,23,27,22,32,37,25,21,14,30,26,15,33,34,9);(6,8,20,34,10,15,29,30,13,21,28,37,12,22,11,23)(7,14,31,9,24,32,17,26,18,19,36,25,16,33,35,27);(6,9,10,26,13,25,12,27)(7,15,16,8,18,22,24,21)(11,19,20,33,29,14,28,32)(17,34,31,23,35,37,36,30)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6258);