Finite Group DatabasebetaFinite Group Databasebeta| order | 32128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C251 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #30631522 |
| commutator, cosocle | C32×C251 | C22 |
| Frattini | C32 | C2×D251 |
| Fitting | Q64×C251 | C2 |
| socle, prime radical | C2×C251 | D32 |
| hypercenter | Q64 | D251 |
| hypocenter | C251 | Q128 |
| LCS | G ▷ C32×C251 ▷ C16×C251 ▷ C8×C251 ▷ C4×C251 ▷ C2×C251 ▷ C251 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 32128 |
Polycyclic presentation:
2^7,251|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2502410 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32128, 2727);