Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C29 · Schur multiplier (H2) = C2×C4 |
| direct product | C29×#353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #99 |
| commutator | C2×C4 | C2×C4×C29 |
| Frattini, cosocle | C2×D4 | C22×C29 |
| prime radical | C29×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C29 ◁ C2×D4×C29 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 232 |
A minimal faithful permutation representation — 2 generators on 37 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)(32,35,37,34)(33,36);(31,34)(32,37)(35,36)377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 34);