Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#2211 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(39,40,41);(4,6)(7,8)(9,17)(10,13)(11,23)(12,26)(14,19)(15,25)(16,22)(18,33)(20,37)(21,34)(24,32)(27,31)(28,35)(29,36)(30,38);(3,4,5,6)(8,13)(14,28)(15,29)(16,26)(17,33)(23,34)(30,32)(31,37);(7,9,10,18)(8,14,13,28)(11,24,21,38)(12,27,22,20)(15,31,29,37)(16,32,26,30)(17,34,33,23)(19,36,35,25)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26566);