Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195972 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(17,18,19,22,20,21,23,24);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(18,21)(19,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=de,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288241);