Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#3347 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
1962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4,7,6)(5,9,10,8)(11,12)(13,20)(14,18)(15,23)(16,19)(17,22)(21,26)(24,25);(3,5)(4,8)(6,9)(7,10)(13,21)(17,25)(20,26)(22,24);(3,6,7,4)(5,9,10,8)(11,13)(12,17)(14,22)(15,24)(16,21)(18,26)(19,25)(20,23);(5,10)(8,9)(11,14,16,15)(12,18,19,23)(13,22,21,24)(17,26,25,20)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54227);