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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(3,5)(9,10)(11,12)(13,14)(15,16)(17,18,23,27)(19,24,29,32)(20,21,25,30)(22,31,28,26);(1,2,6,4)(3,8,5,7)(10,14)(11,15)(17,19,23,29)(18,24,27,32)(20,28,25,22)(21,26,30,31);(1,2,6,4)(3,8,5,7)(9,11,13,15)(10,12,14,16)(17,20,23,25)(18,21,27,30)(19,22,29,28)(24,31,32,26);(1,3,6,5)(2,7,4,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,21)(19,28)(22,29)(23,25)(24,26)(27,30)(31,32);(1,4,6,2)(3,7,5,8)(17,21)(18,25)(19,26)(20,27)(22,24)(23,30)(28,32)(29,31);(1,5,6,3)(2,7,4,8)(17,22,23,28)(18,26,27,31)(19,20,29,25)(21,24,30,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081572);