Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1371 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D8 |
| commutator | C8 | C22×C3×C8 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| socle, prime radical | C23×C3 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(27,28,29);(3,4,5,7,6,8,9,10)(12,16)(13,17)(14,20)(18,26)(19,24)(21,25)(22,23);(11,12,17,23,14,18,21,24,15,19,25,26,20,22,13,16)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4994);