Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#2822 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2)(31,32,33);(8,9)(12,13)(15,16,20,22)(17,23,25,18)(19,21,29,30)(24,28,27,26);(4,6)(7,8,11,9)(10,12,14,13)(15,17,20,25)(16,18,22,23)(19,27,29,24)(21,28,30,26);(3,4)(5,6)(7,9,11,8)(10,12,14,13)(15,18)(16,17)(19,28)(20,23)(21,27)(22,25)(24,30)(26,29);(7,10,11,14)(8,12,9,13)(15,19,20,29)(16,21,22,30)(17,24,25,27)(18,26,23,28)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29671);