Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
1120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(8,9)(10,14)(11,13)(12,15)(16,17,21,25)(18,26,29,30)(19,24,20,23)(22,27,31,28);(8,10,11,15)(9,12,13,14)(18,27)(19,20)(22,30)(23,24)(26,31)(28,29);(4,5,6,7)(16,18,20,27,21,29,19,28)(17,22,24,30,25,31,23,26);(9,13)(12,14)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18529);