Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C3×#496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C22×C3×C42 ◁ G | |
4038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(17,18,19);(3,4);(6,8)(10,13)(11,15)(14,16);(9,10,11,13,12,14,15,16);(5,6,7,8)(10,14)(13,16)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53621);