Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C23 | #1020 |
| commutator, prime radical, cosocle, hypocenter | #73291 | C2 |
| Fitting | #2811 | D3 |
| LCS | G ▷ #73291 | |
| UCS | C1 ◁ C23 | |
207 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
2,3,2^7|a^2=i,c^2=h,d^2=h,e^2=g,f^2=g,b^a=b^2,c^a=cd,d^a=dh,e^a=ef,f^a=fg,c^b=cdh,d^b=c,e^b=efg,f^b=e,d^c=dh,f^c=fghi,e^d=eghi,f^e=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | #72227836, #72227859 |
| maximal normal subgroup of | #72227859 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088604);