Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#39366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(8,9,11,13)(10,14,15,12)(16,17)(18,24)(19,23)(20,21)(22,26)(25,30)(27,28)(29,31);(5,6)(17,21)(22,24)(23,28)(29,30);(4,5,7,6)(8,10,11,15)(9,12,13,14)(16,17)(18,22)(19,23)(20,21)(24,26)(25,29)(27,28)(30,31);(16,18,20,26)(17,22,21,24)(19,25,27,31)(23,29,28,30);(4,6,7,5)(16,19,20,27)(17,23,21,28)(18,25,26,31)(22,29,24,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 32354);