Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#35973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5,8,7)(6,10,11,9)(12,13,16,19)(14,15,18,17);(12,14,16,18)(13,17,19,15)(21,25)(26,32)(27,33)(34,35);(4,6,8,11)(5,9,7,10)(12,13)(14,17)(15,18)(16,19)(20,21)(22,26)(23,27)(24,25)(28,34)(29,32)(30,35)(31,33);(4,7,8,5)(6,10,11,9)(12,15,16,17)(13,18,19,14)(20,22,24,29)(21,26,25,32)(23,30,31,28)(27,35,33,34);(20,23,24,31)(21,27,25,33)(22,28,29,30)(26,34,32,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28961);