Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438880 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,14,18,15,19,13,22)(11,16,21,23,12,17,20,24);(9,10)(11,17)(12,16)(13,18)(14,22)(15,19)(20,24)(21,23);(1,3,4,8)(2,5,6,7)(9,11,14,21,15,12,13,20)(10,16,18,23,19,17,22,24);(9,12,14,20,15,11,13,21)(10,17,18,24,19,16,22,23);(9,13,15,14)(10,18,19,22)(11,20,12,21)(16,23,17,24)Polycyclic presentation:
2^9|a^2=fg,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6353329);