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×#562 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C29 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 928 |
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)(31,34)(32,36)(38,39)(40,44)(41,42)(43,45);(31,34)(35,37)(39,42)(43,44);(30,31)(32,35)(33,34)(36,37)(38,40,41,45)(39,43,42,44);(30,32)(31,35)(33,36)(34,37)(38,39)(40,43)(41,42)(44,45)551 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 243);