Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2189 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1146 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);(3,4)(5,6)(7,8,13,17)(9,18,21,36)(10,15,12,27)(11,23,26,37)(14,29,32,20)(16,22,34,25)(19,35,28,33)(24,30,38,31);(7,9)(8,14)(10,19)(11,24)(12,28)(13,21)(15,30)(16,33)(17,32)(18,23)(20,25)(22,29)(26,38)(27,31)(34,35)(36,37);(4,6)(7,10,13,12)(8,15,17,27)(9,19,21,28)(11,22,26,25)(14,30,32,31)(16,23,34,37)(18,35,36,33)(20,24,29,38)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26503);