Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#41423 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(8,9,12,13)(10,11,15,14)(16,17)(18,20)(19,27)(21,22)(23,25)(24,29)(26,30)(28,31);(9,13)(11,14)(17,22)(18,25)(24,27)(26,31);(4,5,6,7)(9,13)(10,15)(16,18)(17,23)(19,26)(20,22)(21,25)(24,30)(27,28)(29,31);(8,10,12,15)(9,14,13,11)(16,19,21,29)(17,24,22,27)(18,26,25,31)(20,28,23,30);(8,11,12,14)(9,15,13,10)(16,20,21,23)(17,18,22,25)(19,28,29,30)(24,26,27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34411);