Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| 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 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(11,17)(12,19)(15,20)(16,23)(21,22);(1,2,6,5)(3,8,4,7)(9,10,13,15)(11,16,17,22)(12,20,19,14)(18,23,24,21);(1,3,6,4)(2,7,5,8)(10,15)(12,19)(16,22)(18,24);(1,4,6,3)(2,7,5,8)(9,11,13,17)(10,16,15,22)(12,18,19,24)(14,21,20,23);(1,5,6,2)(3,8,4,7)(11,17)(16,22)(18,24)(21,23)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479474);