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 | |
3287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,18,15,11,14,17)(12,20,21,16,23,24,22,19);(1,2,6,5)(3,8,4,7)(9,11,15,10)(12,19,23,16)(13,18,14,17)(20,21,24,22);(1,3,6,4)(2,7,5,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(1,4,6,3)(2,7,5,8)(9,13,15,14)(10,17,11,18)(12,21,23,22)(16,20,19,24);(1,5,6,2)(3,8,4,7)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,f^b=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480603);