Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C23×C3×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| prime radical | C23×C3×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C3×C4 ◁ G | |
11166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 25 points:
(1,2)(23,24,25);(3,4);(5,6);(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21);(7,9,11,17)(8,12,13,21)(10,19,20,16)(14,22,18,15);(8,13)(12,21)(14,18)(15,22)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56070);