Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254275 |
| 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 | #3496 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
960 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)(9,10,15,17)(11,13,19,14)(12,20,21,18)(16,23,24,22);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16,21,24)(14,17)(15,19)(18,23,20,22);(2,6)(3,7)(4,8)(9,12)(10,16)(11,18)(13,23)(14,22)(15,21)(17,24)(19,20);(9,11)(10,13)(12,18)(14,17)(15,19)(16,23)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=ehi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161492);