Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #102 |
| commutator | C8 | C2×C4×C29 |
| Frattini, cosocle | C2×C8 | C22×C29 |
| prime radical | C2×D8×C29 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C2×C8×C29 ◁ G | |
62 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)(30,31,33,37,34,39,44,38)(32,40,42,36,43,35,41,45);(31,35)(32,41)(33,44)(36,37)(38,45)(39,40)(42,43)464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 42);