Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,22,15,12,13,17)(11,18,23,16,21,24,20,19);(2,6)(4,7)(9,11,15,21)(10,16,12,19)(13,23,14,20)(17,24,22,18);(1,2)(3,4)(5,6)(7,8)(9,12)(10,15)(11,19)(13,22)(14,17)(16,21)(18,23)(20,24);(1,3,5,8)(2,7,6,4);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,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, 6524657);