Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C23×C29 |
| prime radical | C29×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ G | |
142 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)(30,31,35,33)(32,38,41,45)(34,42,44,37)(36,43,39,40);(30,32)(31,36)(33,39)(34,43)(35,41)(37,45)(38,42)(40,44);(30,33,35,31)(32,39,41,36)(34,37,44,42)(38,43,45,40)464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 137);