Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2×C4 |
| direct product | C29×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | D4 |
| commutator | C4 | C22×C4×C29 |
| Frattini, cosocle | C2×C4 | C23×C29 |
| socle | C22×C29 | C2×D4 |
| prime radical | C23×C29 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C22×C4×C29 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 49 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)(30,31,32,33)(34,35)(36,43)(37,41)(38,47)(39,42)(40,45)(44,48)(46,49);(34,36,38,45,39,46,37,44)(35,40,41,43,42,48,47,49);(34,37,39,38)(35,41,42,47)(36,44,46,45)(40,43,48,49)812 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 108);