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 | #18380 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
981 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)(3,4)(5,7)(6,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,20)(19,24)(21,23);(1,3,6,7)(2,4,8,5)(9,10)(11,18,19,24)(12,22)(13,16)(14,17)(15,21,23,20);(2,3)(4,5)(7,8)(9,11)(10,15)(12,21)(13,20)(14,19)(16,24)(17,23)(18,22);(9,10)(11,15)(12,22)(13,16)(14,17)(18,21)(19,23)(20,24)Polycyclic presentation:
2^9|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, 161480);