Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C3 |
| direct product | Q8×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×H(3,𝔽3) |
| commutator, Frattini, prime radical, cosocle | C2×C32 | C22×C32 |
| LCS | G ▷ C2×C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C32×Q8 ◁ G | |
78 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)(5,8,7,6)(9,10,13,12,16,22,20,15,26)(11,17,21,19,14,23,30,25,33)(18,28,31,29,27,32,35,24,34);(1,5,3,7)(2,6,4,8)(9,11,18,20,30,35,12,19,29)(10,14,24,15,17,27,16,25,28)(13,21,31,26,33,34,22,23,32)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 231);