Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24272 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(7,14)(10,15)(12,20)(16,17);(2,4)(5,6,8,18,9,11,19,13)(7,15,17,12,14,10,16,20);(6,10)(7,16)(8,19)(11,15)(12,18)(13,20)(14,17);(2,4)Polycyclic presentation:
2^9|b^2=fh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72252748, #72252775, #72252778, … |
| maximal normal subgroup of | #72252748, #72252775, #72252778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409006);