Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C3×#60058 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C24 |
| commutator | C2 | C23×C3×C42 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C24×C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(8,9,12,16,13,17,23,29)(10,14,20,26,19,25,32,36)(11,15,22,28,21,27,33,37)(18,24,31,35,30,34,38,39);(4,5,6,7)(8,10)(9,14)(11,18)(12,20)(13,19)(15,24)(16,26)(17,25)(21,30)(22,31)(23,32)(27,34)(28,35)(29,36)(33,38)(37,39);(8,11)(9,15)(10,18)(12,22)(13,21)(14,24)(16,28)(17,27)(19,30)(20,31)(23,33)(25,34)(26,35)(29,37)(32,38)(36,39);(11,21)(15,27)(18,30)(22,33)(24,34)(28,37)(31,38)(35,39);(10,19)(14,25)(18,30)(20,32)(24,34)(26,36)(31,38)(35,39)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53046);