Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#2689 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2526 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);(7,8,11,13)(9,14,10,12)(15,16)(17,20)(18,22)(19,21);(4,6)(7,9,11,10)(8,12,13,14)(15,17,19,22)(16,20,21,18);(3,4)(5,6)(7,10,11,9)(8,12,13,14)(15,17)(16,18)(19,22)(20,21);(7,8)(9,14)(10,12)(11,13)(15,18)(16,17)(19,20)(21,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 45097);