Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#564 |
| 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 | |
172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 4 generators on 61 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(32,38)(33,41)(35,39)(36,43)(46,47,50,53)(48,51,56,54)(49,57,58,52)(55,60,61,59);(30,31,34,37)(32,39,38,35)(33,36,41,43)(40,45,44,42)(46,47)(48,54)(49,57)(50,53)(51,56)(52,58)(55,59)(60,61);(30,32,34,38)(31,35,37,39)(33,40,41,44)(36,42,43,45)(46,48)(47,51)(49,55)(50,56)(52,59)(53,54)(57,60)(58,61);(30,33)(31,36)(32,40)(34,41)(35,42)(37,43)(38,44)(39,45)(46,49,50,58)(47,52,53,57)(48,55,56,61)(51,59,54,60)551 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 245);