Finite Group DatabasebetaFinite Group Databasebeta| order | 32128 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | #564×C251 ⋊ C2 |
| Sylow | C251 ⋊ #3567 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D251 |
| commutator, socle, cosocle | C22×C251 | C25 |
| Fitting | #564×C251 | C2 |
| hypercenter | #564 | D251 |
| hypocenter | C251 | #3567 |
| LCS | G ▷ C22×C251 ▷ C251 | |
| UCS | C1 ◁ C22 ◁ #564 | |
950 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 449 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 16064 |
Polycyclic presentation:
2^7,251|c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^250,c^b=cg,d^b=df,h^b=h^250,h^c=h^250,h^e=h^2502413 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32128, 19200);