Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22×C4 |
| direct product | C29×#410 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | C22⋊C4 |
| commutator | C22 | C22×C4×C29 |
| Frattini, cosocle | C23 | C23×C29 |
| socle | C2×C29 | C2×C22⋊C4 |
| prime radical | C2×#41×C29 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C22×C4×C29 ◁ G | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 3 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)(32,38,40,35)(33,36)(34,37)(39,44,45,42)(41,43);(30,32)(31,35)(33,39)(34,40)(36,42)(37,38)(41,45)(43,44);(30,33,34,41)(31,36,37,43)(32,39,40,45)(35,42,38,44)638 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 91);