Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#2398 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| socle | C24×C3 | C24 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
3246 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)(15,16,18,20)(17,21,22,19);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17,18,22)(16,19,20,21);(4,6)(16,20)(19,21);(7,10,11,9)(8,12,13,14)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27533);