Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348967 |
| 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 | C42 | C25 |
| prime radical | #60561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,20)(12,23)(13,19)(14,18)(15,17)(16,21)(22,24);(1,2,5,7)(3,6,4,8)(9,11,15,12)(10,16,17,24)(13,22,14,21)(18,23,19,20);(1,3,5,4)(2,6,7,8)(9,12,15,11)(10,16,17,24)(13,21,14,22)(18,23,19,20);(10,17)(16,24)(18,19)(20,23);(1,3,5,4)(2,6,7,8)(10,17)(16,24)(18,19)(20,23)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256209);