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×#41409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2434 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)(11,13)(12,15)(16,17,21,25)(18,22,26,31)(19,23,29,28)(20,24,30,27);(8,9,11,13)(10,12,15,14)(18,26)(20,30)(22,31)(24,27);(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,18)(17,22)(19,24)(20,28)(21,26)(23,30)(25,31)(27,29);(8,10,11,15)(9,12,13,14)(16,19)(17,23)(18,27)(20,22)(21,29)(24,26)(25,28)(30,31);(16,20)(17,24)(18,28)(19,22)(21,30)(23,26)(25,27)(29,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34397);