Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,10,11)(7,9,12,8)(13,14,16,21,18,15,20,24,19,23,17,22,27,26,28,25);(2,4)(5,7,9,11,10,12,8,6)(13,15,17,25,18,23,28,21,19,26,16,24,27,14,20,22);(5,8,10,9)(6,7,11,12)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fg,c^b=cg176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34014);