Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#13268 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C3×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C8 |
| prime radical | C23×C3 | C22×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,15,19,16,20,23,26,17,14,18,21,24,22,25,27);(5,8)(9,10)(12,14,15,19,16,22,23,26,17,13,18,21,24,20,25,27);(4,6,7,11)(5,9,8,10)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6256);