Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501868 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1771 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,13,8,14)(7,9)(10,12)(15,20,18,19)(16,17);(1,2)(3,4)(5,6,10,15,11,8,12,18)(7,13,9,19,16,14,17,20);(2,4)(5,7,10,9,11,16,12,17)(6,14,15,20,8,13,18,19);(5,8,11,6)(7,13,16,14)(9,19,17,20)(10,18,12,15)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,d^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,e^b=eh,h^b=hi,e^d=ehi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72250290, #72252778, #72252782, … |
| maximal normal subgroup of | #72250290, #72252778, #72252782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411368);