Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 15552 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C9 |
| abelian | C8×C92 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C32×C4 | C2×C32 |
| socle, prime radical | C2×C32 | C32×C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17);(18,19,20,21,22,23,24,25,26)648 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 32);