Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61667 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(13,19)(14,22)(16,18)(20,27)(23,26)(24,25);(4,6)(5,8)(7,11)(9,10)(12,13,17,19)(14,20,24,26)(15,18,21,16)(22,23,25,27);(12,14)(13,20)(15,23)(16,22)(17,24)(18,25)(19,26)(21,27);(6,11)(8,10)(12,15,17,21)(13,16,19,18)(14,23,24,27)(20,22,26,25);(5,9)(8,10)(12,16,17,18)(13,21,19,15)(14,22,24,25)(20,27,26,23)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54655);