Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#13108 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #387 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1917 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
246 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)(6,7,11,18,12,16,30,35,13,17,31,19,10,26,28,22)(8,9,15,25,20,14,32,33,21,24,34,29,27,23,37,36);(6,8,11,15,12,20,30,32,13,21,31,34,10,27,28,37)(7,14,18,33,16,24,35,29,17,23,19,36,26,9,22,25);(6,9,10,23,13,24,12,14)(7,15,16,32,17,34,26,37)(8,19,20,22,21,18,27,35)(11,29,28,33,31,25,30,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6096);