Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #386 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,7,8)(10,13,14,16)(17,18,21,23,20,19,22,24);(1,2,6,5,4,7,3,8)(11,15)(13,16)(17,19,21,24,20,18,22,23);(9,10,12,14)(11,16,15,13)(18,19)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=h,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,c^b=ci,d^b=dg,e^b=eh,f^b=fg,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, 6721);