Finite Group DatabasebetaFinite Group Databasebeta| order | 32128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #399×C251 ⋊ C2 |
| Sylow | C251 ⋊ #2842 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D251 |
| commutator, socle, cosocle | C23×C251 | C24 |
| Fitting | #399×C251 | C2 |
| prime radical | C24×D251 | C22 |
| hypercenter | #399 | D251 |
| hypocenter | C251 | #2842 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #399 | |
548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 8032 |
Polycyclic presentation:
2^7,251|b^2=eg,c^2=g,b^a=be,c^a=cf,h^a=h^250,c^b=cg,d^b=def,d^c=de2776 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32128, 12775);