Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
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×#120 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,20,23,21,19,22,24);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,15,12,16);(2,6)(5,8)(9,11,13,12)(10,15,14,16);(10,14)(15,16);(9,12,13,11)(10,16,14,15)(17,19,20,24,21,18,22,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6309602);