Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Q8⋊C64 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | D4 |
| commutator | C4 | C2×C64 |
| Frattini, cosocle | C4×C32 | C22 |
| socle, prime radical | C22 | C22⋊C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 72 points:
(2,5)(3,7)(6,8)(9,10,11,16,12,17,21,31,13,18,22,32,25,35,41,51,14,19,23,33,26,36,42,52,28,38,44,54,47,57,61,66,15,20,24,34,27,37,43,53,29,39,45,55,48,58,62,67,30,40,46,56,49,59,63,68,50,60,64,69,65,70,71,72);(1,2,4,6)(3,8,7,5)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=e,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,c^b=ce224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1812);