Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C29 |
| prime radical | C29×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ G | |
216 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)(30,31)(32,36)(33,35)(34,37)(40,44)(42,45);(30,31,33,35)(32,34,37,36)(38,39,41,43)(40,42,44,45);(30,31)(32,34)(33,35)(36,37)(38,40)(39,42)(41,44)(43,45);(30,32,33,37)(31,34,35,36)638 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 223);