Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34491 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(4,5,7,6)(9,12)(10,11)(17,21)(19,27)(22,24)(25,31);(8,9,13,12)(10,15,11,14)(16,17)(18,24)(19,23)(20,21)(22,26)(25,30)(27,28)(29,31);(8,10,13,11)(9,14,12,15);(4,6,7,5)(8,11,13,10)(9,15,12,14)(16,18,20,26)(17,22,21,24)(19,25,27,31)(23,29,28,30);(8,12,13,9)(10,15,11,14)(16,19,20,27)(17,23,21,28)(18,25,26,31)(22,29,24,30)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27479);