Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9);(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32);(19,25)(22,28)(26,31)(29,32);(2,7)(6,15)(8,12)(9,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=di110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252136);