Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | Q82×C29 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C29 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C29 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ G | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4)(5,8,7,6)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(1,5,3,7)(2,6,4,8);(9,10,11,12)(13,16,15,14);(9,13,11,15)(10,14,12,16)725 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 239);