Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C16 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×C4⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C8 | C22 |
| commutator | C2 | C23×C3×C16 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle, prime radical | C24×C3 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C8 ◁ G | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(25,26,27);(3,4);(6,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(5,6,7,8)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26382);