Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C3×#2820 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(7,8,10,12)(9,11,13,14)(16,19)(20,21);(4,6)(9,13)(11,14)(15,16,18,19)(17,20,22,21);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,21)(18,19)(20,22);(7,9)(8,11)(10,13)(12,14)(15,17,18,22)(16,20,19,21)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29669);