Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34829 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,24,22,26,19,21)(20,25,29,28,30,32,27,31);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,23)(18,24,26,21)(20,27,30,29)(25,28,32,31);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,20,22,30)(18,25,26,32)(19,27,23,29)(21,31,24,28);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,21,19,26,22,24,23,18)(20,28,27,25,30,31,29,32);(9,12,13,11)(10,15,14,16)(17,19,22,23)(18,24,26,21)(20,29,30,27)(25,31,32,28)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6330944);