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×#560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C29 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 4 generators on 45 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)(30,31)(32,39)(33,38)(34,36)(35,41)(37,43)(40,44)(42,45);(30,32)(31,35)(33,41)(34,42)(36,44)(37,45)(38,39)(40,43);(31,36)(32,40)(35,39)(37,38)(41,42)(44,45);(31,37)(32,41)(35,44)(36,38)(39,45)(40,42)551 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 241);