Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437935 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,11)(12,18,21,16)(13,19,15,17)(20,23,24,22);(9,11,13,19,14,10,15,17)(12,20,22,18,21,24,23,16);(1,3,5,4)(2,6,7,8)(9,12,14,21)(10,16,11,18)(13,22,15,23)(17,20,19,24);(1,4,5,3)(2,6,7,8)(9,11,14,10)(12,18,21,16)(13,19,15,17)(20,22,24,23);(12,21)(16,18)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359427);