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 | |
4835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,7,8,4)(10,11)(14,15);(1,3)(2,4)(5,7)(6,8)(9,10,13,11)(12,14,16,15)(18,23)(19,25)(24,31)(26,32);(9,10,13,11)(12,14,16,15)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27);(2,5)(3,8)(9,10,13,11)(12,15,16,14)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32);(1,4)(2,3)(5,8)(6,7)(9,11,13,10)(12,14,16,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30);(1,5,6,2)(3,7,8,4)(9,12)(10,14)(11,15)(13,16)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8602517);