Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#346 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C29 | C22 |
| commutator | C2 | C4×C8×C29 |
| Frattini, cosocle | C2×C8 | C22×C29 |
| socle, prime radical | C22×C29 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C29 ◁ G | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 2 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)(34,35,36,40,37,41,42,47,38,39,43,45,44,46,48,49);(30,31,32,33)(35,39)(40,45)(41,46)(47,49)1160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 27);