Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#471 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×AGL(1,ℤ/8) | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(11,17)(12,19)(15,22)(16,21)(20,23);(2,6)(4,7)(9,10,12,14,13,16,19,21)(11,15,18,20,17,23,24,22);(1,2)(3,4)(5,6)(7,8)(9,11,12,18,13,17,19,24)(10,15,14,20,16,23,21,22);(1,3,5,8)(2,7,6,4);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6525556);