Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #24262 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,3)(4,8,5,7)(10,15)(12,13)(16,18)(17,24)(19,23)(20,22);(1,3,6,2)(4,7,5,8)(10,16,18,15)(17,24,19,23);(2,7)(3,8)(4,5)(9,10)(11,19)(12,16)(13,15)(14,18)(17,21)(20,24)(22,23);(1,4,6,5)(2,8,3,7)(9,11,14,21)(10,17,18,19)(12,22,13,20)(15,23,16,24)Polycyclic presentation:
2^9|a^2=hi,b^2=ehi,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gi,f^d=fh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161661);