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×#41404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2586 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)(10,14)(11,12)(13,15)(16,17,21,22)(18,24,27,23)(19,28,30,20)(25,31,29,26);(9,12)(13,14)(17,22)(20,28)(23,24)(26,31);(4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,18,21,27)(17,23,22,24)(19,29,30,25)(20,26,28,31);(8,10)(9,13)(11,15)(12,14)(16,19)(17,20)(18,25)(21,30)(22,28)(23,31)(24,26)(27,29);(16,20)(17,19)(18,26)(21,28)(22,30)(23,29)(24,25)(27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34392);