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 | |
4673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,14)(13,16)(18,23)(21,29)(24,30)(28,32);(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,13,16,15)(20,27)(24,30)(26,31)(28,32);(2,3)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32);(2,3)(4,8)(9,11,12,16)(10,13,14,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32);(1,3,5,2)(4,7,8,6)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32);(1,4)(2,6)(3,7)(5,8)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012504);