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×#541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C29 |
| prime radical | C23×C29 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 4 generators on 53 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)(32,36)(34,37)(38,39,42,45)(40,43,48,46)(41,49,50,44)(47,52,53,51);(30,31,33,35)(32,34,36,37)(38,39)(40,46)(41,44)(42,45)(43,48)(47,52)(49,50)(51,53);(30,32)(31,34)(33,36)(35,37)(38,40)(39,43)(41,47)(42,48)(44,51)(45,46)(49,52)(50,53);(38,41,42,50)(39,44,45,49)(40,47,48,53)(43,51,46,52)638 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 222);