Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#37260 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2066 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)(5,7)(8,9,11,12)(10,14,15,13)(16,17,22,21)(18,25,20,23)(19,24,29,28)(26,31,27,30);(9,12)(13,14)(16,18,22,20)(17,23,21,25)(19,27,29,26)(24,31,28,30);(4,5)(6,7)(16,19,22,29)(17,24,21,28)(18,26,20,27)(23,30,25,31);(8,9)(10,13)(11,12)(14,15)(16,20,22,18)(17,25,21,23)(19,27,29,26)(24,31,28,30);(8,10,11,15)(9,13,12,14)(16,21,22,17)(18,25,20,23)(19,28,29,24)(26,31,27,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30248);