Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C32 |
| Fitting | #2258 | D3 |
| socle | C2 | A4⋊C32 |
| prime radical | #41⋊C3 | C16 |
| cosocle | #2258⋊C3 | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C32 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
2^5,3,2^3|a^2=b,b^2=c,c^2=d,d^2=e,e^2=i,g^2=i,h^2=i,f^a=f^2,g^a=gh,h^a=hi,g^f=h,h^f=ghi,h^g=hi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085352);