Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C24×C8 |
| socle | C23 | C24×C4 |
| prime radical | C2×#555 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,23,21,19,22,24);(1,3,6,2)(4,7,5,8)(11,16)(13,15);(9,11,12,16)(10,13,14,15);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,19,20,24,21,18,22,23);(1,5,6,4)(2,7,3,8)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6331768);