Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254371 |
| 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 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3)(4,6,8,7)(10,15)(12,13)(16,18)(17,23)(19,21)(22,24);(1,3,5,2)(4,7,8,6)(10,16,18,15)(17,23,24,22);(2,6)(3,7)(4,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,22)(20,24)(21,23);(9,11)(10,17)(12,21)(13,19)(14,20)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=hi,b^2=ehi,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,e^c=eh,f^c=fi,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161612);