Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13197 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #121 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1954 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
424 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)(6,7,11,18,12,17,28,33,13,19,27,15,10,16,9,26)(8,20,24,36,22,32,21,35,25,34,29,37,23,14,30,31);(6,8,9,21,10,23,27,24,13,25,28,30,12,22,11,29)(7,14,26,36,16,34,15,31,19,32,33,37,17,20,18,35);(6,9,10,27,13,28,12,11)(7,15,17,26,19,18,16,33)(8,21,23,24,25,30,22,29)(14,31,20,36,32,35,34,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6185);