Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #386 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,12,15,13,11,14,16)(17,18,21,24,22,19,20,23);(1,2,6,8,4,7,3,5)(9,11,12,16,13,10,14,15)(17,19,22,18)(20,23,21,24);(2,7)(5,8)(17,18,22,19)(20,24,21,23)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7062);