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×#41413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2380 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)(16,17,21,25)(18,26,27,22)(19,23,29,31)(20,24,30,28);(8,9,11,12)(10,13,14,15)(18,27)(20,30)(22,26)(24,28);(4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,18,21,27)(17,22,25,26)(19,28,29,24)(20,23,30,31);(8,10,11,14)(9,13,12,15)(16,19)(17,23)(18,24)(20,22)(21,29)(25,31)(26,30)(27,28);(16,20)(17,24)(18,23)(19,22)(21,30)(25,28)(26,29)(27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34401);