Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | D8×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D4×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32×C4 | C22×C32 |
| prime radical | D8×H(3,𝔽3) | C3 |
| LCS | G ▷ C32×C4 ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C32×C4 ◁ G | |
154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8)(9,10,13)(11,17,21)(12,16,22)(14,23,19)(15,26,20)(18,28,31)(24,34,35)(25,33,30)(27,32,29);(2,8)(3,7)(4,6)(9,11,18,12,19,29,20,30,35)(10,14,24,16,25,28,15,17,27)(13,21,31,22,23,32,26,33,34)119 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 609);