Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13201 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #390 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1955 |
| prime radical | C5×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
420 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)(6,7,11,17,12,16,31,35,13,18,32,21,10,15,20,26)(8,19,23,37,22,29,34,36,24,27,14,33,30,9,25,28);(6,8,20,34,10,30,32,23,13,24,31,25,12,22,11,14)(7,9,26,37,15,27,21,28,18,29,35,33,16,19,17,36);(6,9,10,27,13,29,12,19)(7,14,16,25,18,23,15,34)(8,21,22,26,24,17,30,35)(11,28,20,33,32,36,31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6189);