Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C22×C3×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | D4 |
| commutator | C4 | C23×C3×C8 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C23×C3 | C4⋊C8 |
| prime radical | C24×C3 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ C23×C3×C8 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(21,22,23);(3,4);(5,6,9,14,10,13,18,20,11,15,17,12,8,19,7,16);(5,7,8,17,11,18,10,9)(6,12,13,16,15,14,19,20)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26387);