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, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(6,10)(11,12)(14,17)(15,19)(18,20);(5,6,9,10)(7,11,8,12);(5,7,9,8)(6,11,10,12)(13,14,15,17,16,18,19,20);(5,8,9,7)(6,11,10,12);(14,18)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287533);