Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#42994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2264 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)(9,12)(10,14)(16,17)(18,23)(19,27)(20,25)(21,22)(24,30)(26,29)(28,31);(8,9,11,12)(10,13,14,15)(17,22)(20,23)(24,27)(29,31);(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,18,21,25)(17,23,22,20)(19,28,30,26)(24,29,27,31);(8,10)(9,13)(11,14)(12,15)(16,19,21,30)(17,24,22,27)(18,26,25,28)(20,29,23,31);(16,20,21,23)(17,25,22,18)(19,29,30,31)(24,28,27,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35982);