Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,8,5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,28)(21,24)(22,25)(23,30)(27,29)(31,32);(1,3,6,5)(2,7,4,8)(9,11,12,16)(10,13,14,15);(1,2,6,4)(3,8,5,7)(17,19,22,24)(18,21,25,26)(20,27,30,31)(23,29,28,32);(17,20)(18,23)(19,27)(21,29)(22,30)(24,31)(25,28)(26,32);(1,4,6,2)(3,8,5,7)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,30)(31,32);(1,5,6,3)(2,7,4,8)(17,21)(18,24)(19,25)(20,29)(22,26)(23,31)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561824);