Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437995 |
| 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 | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,24)(20,23);(9,11)(10,14)(12,19,20,17)(13,18)(15,21,23,24)(16,22);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(10,16)(14,22)(15,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353002);