Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C8 |
| direct product | C3×#7529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1408 |
| commutator | C4×C8 | C2×C3×C4 |
| Frattini, cosocle | #493 | C22×C3 |
| prime radical | C3×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×#493 ◁ G | |
186 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5)(6,11,14,10)(7,12,8,17)(9,13)(15,19,16,18);(5,10)(6,14)(7,8)(11,12)(13,19)(15,16)(17,18)75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 517);