Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C29 · Schur multiplier (H2) = C1 |
| direct product | C29×#365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #106 |
| commutator | C8 | C2×C4×C29 |
| Frattini, cosocle | C2×C8 | C22×C29 |
| prime radical | C22×C29 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C2×C8×C29 ◁ G | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 2 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)(31,35,36,41)(32,38)(33,40)(37,44,43,42)(39,45);(30,31,32,35,33,37,38,42,34,36,39,41,40,43,45,44)464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 46);