Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#1445 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | C23 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ G | |
2942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(23,24,25);(7,8,9,10)(12,14)(16,19)(20,21);(4,6)(8,10)(11,12)(13,14)(15,16,18,19)(17,21,22,20);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,18,22)(16,20,19,21)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6833);