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×#490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C29 |
| prime radical | C29×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 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)(32,35)(33,34)(36,37)(39,44)(40,45)(41,42);(31,34)(35,37)(38,39,41,44,43,45,42,40);(30,32,33,36)(31,35,34,37)(38,40,41,39,43,44,42,45)464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 171);