Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#13277 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D8 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(5,7)(8,9)(10,14)(11,12)(13,15)(16,17,20,25,28,31,27,22,19,23,18,24,21,26,29,30);(4,5,6,7)(9,12)(13,14)(17,22)(18,27)(20,29)(21,28)(23,30)(24,26)(25,31);(8,10,11,15)(9,13,12,14)(17,23)(22,30)(24,25)(26,31)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6265);