Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,7,5,3,8,6)(10,14,15,16);(1,3)(2,5)(4,6)(7,8)(9,10,12,14,13,15,11,16);(1,2)(3,5)(4,7)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=eh,f^b=fhi,d^c=dh,e^c=eh104 conjugacy classes — view the full character table.
| maximal subgroup of | #1042, #72226262, #72226265, … |
| maximal normal subgroup of | #1042, #72226262, #72226265, … |
| maximal quotient of | #72239502 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14253);