Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C4 |
| direct product | C29×#491 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C29 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C29 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 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)(30,31)(32,35)(33,34)(36,37)(38,39,44,49)(40,46,51,52)(41,45,50,53)(42,48,43,47);(31,34)(35,37)(38,40,42,41,44,51,43,50)(39,45,47,46,49,53,48,52);(30,32,33,36)(31,35,34,37)(38,41,42,51,44,50,43,40)(39,46,47,53,49,52,48,45)464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 172);