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 | |
4888 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)(3,4)(5,6)(7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(1,3,6,7)(2,4,5,8)(10,14)(11,15)(18,22)(23,29)(24,30)(31,32);(1,4,6,8)(2,7,5,3)(9,11)(10,12)(13,15)(14,16)(19,25)(23,29)(26,27)(31,32);(1,2,6,5)(3,8,7,4)(9,12)(10,11)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(1,4)(2,3)(5,7)(6,8)(17,19,21,25)(18,23,22,29)(20,27,28,26)(24,32,30,31);(1,5,6,2)(3,8,7,4)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9983126);