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 | #61321 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,6,8,7)(10,14)(11,17)(12,20)(16,22)(18,19)(21,23);(2,4)(6,7)(9,10,12,14,13,16,20,22)(11,15,18,21,19,24,17,23);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,21)(16,24)(17,20)(22,23);(1,3,5,8)(2,6,4,7);(1,4,5,2)(3,6,8,7)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6534565);