Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C8 · Schur multiplier (H2) = C210 |
| direct product | C22×C3×C8×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C8 | C22 |
| commutator | C2 | C24×C3×C8 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C24×C3 | C22×C4 |
| prime radical | C23×C3×D4 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C8 ◁ G | |
3694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(17,18,19);(3,4);(5,6,7,8,9,10,11,12);(13,14,15,16);(14,16)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53118);