Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #118356 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,14)(12,19,21,15)(13,16)(17,24,20,22)(18,23);(1,2,3,5,4,6,7,8)(9,11)(10,14)(12,20,21,17)(13,18)(15,22,19,24)(16,23);(2,6)(3,7)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25552);