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 | |
4561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,5,4,7)(17,18,22,23)(19,26,25,21)(20,24,28,32)(27,30,29,31);(1,3,6,4)(2,7,8,5)(9,10)(11,15)(12,14)(13,16)(18,23)(19,25)(20,28)(30,31);(9,11)(10,13)(12,16)(14,15)(18,23)(19,25)(24,32)(27,29);(2,8)(3,4)(10,14)(13,15)(17,19,22,25)(18,21,23,26)(20,29,28,27)(24,30,32,31);(1,4,6,3)(2,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,20,22,28)(18,24,23,32)(19,27,25,29)(21,31,26,30);(1,5,6,7)(2,4,8,3)(9,11)(10,15)(12,16)(13,14)(17,21)(18,25)(19,23)(20,30)(22,26)(24,27)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10127586);