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 | |
4992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,6,7,4)(18,23)(19,27)(25,31)(26,28);(1,2,5,8)(3,4,7,6)(9,10)(11,12)(13,15)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16);(3,7)(4,6)(9,12,13,14)(10,11,15,16)(17,19,22,27)(18,24,23,29)(20,26,30,28)(21,31,32,25);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20,22,30)(18,25,23,31)(19,28,27,26)(21,24,32,29);(1,4,5,6)(2,7,8,3)(9,10)(11,14)(12,16)(13,15)(17,21,22,32)(18,26,23,28)(19,25,27,31)(20,29,30,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131389);