Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
2648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,14)(12,15,16,13);(2,4)(9,10)(11,14,16,13)(12,15);(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=dh,f^b=fhi,h^b=hi,f^d=fi,g^d=gi,g^f=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225750, #72226368, #72258001, … |
| maximal normal subgroup of | #72225750, #72226368, #72258001, … |
| maximal quotient of | #72219296, #72219297, #72258221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53765);