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×#24202 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5)(6,7)(8,9)(10,14)(11,12)(13,15)(18,26)(19,20)(22,30)(23,24)(27,31)(28,29);(9,12)(13,14)(16,17,21,25)(18,27,28,30)(19,24,20,23)(22,26,31,29);(5,7)(16,18,21,28)(17,22,25,31)(19,29,20,26)(23,27,24,30);(8,10,11,15)(9,13,12,14)(18,28)(22,31)(26,29)(27,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17190);