Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C92 · Schur multiplier (H2) = C9 |
| direct product | Q8×C92 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C92 | C22 |
| commutator | C2 | C22×C92 |
| Frattini, socle, prime radical, cosocle | C2×C32 | C22×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C92 ◁ G | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17);(1,5,3,7)(2,6,4,8)(18,19,20,21,22,23,24,25,26)405 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 223);