Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×C128 ⋊ C2 |
| Sylow | C3 ⋊ M256 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C64 | C2×D3 |
| commutator, socle | C2×C3 | C2×C64 |
| Fitting | C3×C128 | C2 |
| prime radical | C2×D3 | C64 |
| cosocle | C3×C64 | C22 |
| hypercenter | C128 | D3 |
| hypocenter | C3 | M256 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C64 ◁ C128 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bh,i^b=i^2288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 57478);