Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #2671 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,24,26,23,27,21,32)(19,28,29,25,30,22,20,31);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19,23,30)(18,22,27,28)(20,24,29,21)(25,26,31,32);(9,12,13,11)(10,14,15,16)(17,20,23,29)(18,25,27,31)(19,21,30,24)(22,32,28,26);(1,3,4,7)(2,6,5,8)(17,21,23,24)(18,26,27,32)(19,29,30,20)(22,25,28,31);(2,5)(6,8)(17,22,24,31,23,28,21,25)(18,20,26,19,27,29,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6312340);