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×#37771 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2656 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2216 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)(10,12,15,14)(16,17,21,22)(18,25,23,19)(20,24,28,29)(26,31,30,27);(5,7)(8,9,11,13)(10,14,15,12)(17,22)(18,23)(24,29)(26,30);(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,18,21,23)(17,19,22,25)(20,26,28,30)(24,27,29,31);(16,19)(17,23)(18,22)(20,27)(21,25)(24,30)(26,29)(28,31);(4,5)(6,7)(8,10,11,15)(9,12,13,14)(16,20,21,28)(17,24,22,29)(18,26,23,30)(19,27,25,31)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30759);