Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#39880 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(4,5,6,7)(16,17,21,20)(18,24,26,22)(19,27,28,23)(25,29,31,30);(9,13)(11,14)(17,20)(22,24)(23,27)(29,30);(4,5)(6,7)(8,9)(10,14)(11,15)(12,13)(16,18,21,26)(17,22,20,24)(19,25,28,31)(23,29,27,30);(8,10,12,15)(9,14,13,11)(16,19)(17,23)(18,25)(20,27)(21,28)(22,29)(24,30)(26,31);(8,11,12,14)(9,15,13,10)(16,20,21,17)(18,24,26,22)(19,27,28,23)(25,30,31,29)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 32868);