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 | #59834 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,8,4,7)(10,14)(15,16)(17,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,29)(30,32);(9,10)(11,15)(12,16)(13,14)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,32,30,31);(1,3,6,4)(2,7,5,8)(9,11,13,12)(10,15,14,16)(18,20)(21,29)(23,25)(26,30);(1,2,6,5)(3,7,4,8)(17,19,22,27)(18,23,20,25)(21,30,29,26)(24,32,28,31);(1,4,6,3)(2,7,5,8)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,32,30,31);(1,5,6,2)(3,7,4,8)(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,31)(25,32)(27,30)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995161);