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 | |
4949 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)(11,12)(13,14)(15,16)(17,18,22,23)(19,20,27,24)(21,25,30,31)(26,28,29,32);(1,3,7,6)(2,4,8,5)(10,14)(11,15)(18,23)(20,24)(25,31)(28,32);(1,4,7,5)(2,6,8,3)(9,11,13,15)(10,12,14,16);(1,4,7,5)(2,3,8,6)(9,12)(10,15)(11,14)(13,16)(17,19,22,27)(18,24,23,20)(21,29,30,26)(25,28,31,32);(1,5,7,4)(2,3,8,6)(17,20,22,24)(18,19,23,27)(21,28,30,32)(25,26,31,29);(1,6,7,3)(2,4,8,5)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352144);