Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106654 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,6,4,5,7,8)(9,10)(11,15)(12,14)(13,16);(2,5)(3,7)(9,11)(10,13,14,15)(12,16);(2,5)(3,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,f^b=fh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232717, #72232753, #72232754, … |
| maximal normal subgroup of | #72232717, #72232753, #72232754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13856);