Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25848 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(8,9,12,15)(10,14,11,13)(16,17)(18,23)(19,29)(20,24)(21,25)(22,26)(27,30)(28,31);(8,10,12,11)(9,13,15,14)(16,18)(17,23)(19,20)(21,27)(22,28)(24,29)(25,30)(26,31);(4,5,6,7)(16,19,20,18,22,27,21,28)(17,23,25,29,26,31,24,30);(8,11,12,10)(9,14,15,13)(16,20,22,21)(17,24,26,25)(18,19,28,27)(23,29,31,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18836);