Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C29 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D29 |
| commutator, socle, cosocle | C22×C29 | C24 |
| Fitting | C29×#123 | C2 |
| prime radical | C2×D4×C29 | C22 |
| hypercenter | #123 | D29 |
| hypocenter | C29 | #563 |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ #123 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 928 |
Polycyclic presentation:
2^6,29|a^2=e,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=def,g^a=g^28,c^b=cef,d^b=de215 conjugacy classes — view the full character table.
| maximal subgroup of | #16821503, #16821506, D29×#563, … |
| maximal normal subgroup of | #16821503, #16821506, D29×#563, … |
| maximal quotient of | #16822635, #16822640, #16822659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 1302);