Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #62624 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C8 |
| Frattini | C8 | C23×A4 |
| Fitting | #62624 | C3 |
| socle | C2 | C22×C8×A4 |
| prime radical | #5945 | C4 |
| cosocle | #131 | C23×C3 |
| hypercenter | #504 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C16 ◁ #504 | |
473 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=g,g^2=h,h^2=i,c^a=cefh,d^a=e,e^a=defhi,f^c=fi,e^d=ei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084939);