Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,13)(11,19,20,17)(12,14)(16,18)(21,24,22,23);(1,3,4,7)(2,5,6,8)(10,16)(13,18)(17,19)(23,24);(2,6)(5,8)(9,10,14,18,15,16,12,13)(11,17,21,24,20,19,22,23);(9,11)(10,17)(12,22)(13,23)(14,21)(15,20)(16,19)(18,24)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=eh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=ghi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419431);