Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C24×C3×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ C24×C3×C4 ◁ G | |
11166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2)(25,26,27);(3,4);(5,6);(7,8);(9,10,11,12,13,14,15,16)(17,24,23,22,21,20,19,18);(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56067);