Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499837 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3404 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,18,24)(12,20,13,21)(16,19,17,23);(2,5)(3,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,23)(17,19)(18,24);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,24,23);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,21,22,20)(15,23,24,19)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419922);