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 |
| prime radical | #35564 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,12)(13,14)(15,16);(1,3,5,8)(2,6,7,4)(10,14)(11,15)(17,18,22,21)(19,23,25,27)(20,24,28,30)(26,31,29,32);(1,2)(3,4)(5,7)(6,8)(9,11,13,15)(10,12,14,16)(18,21)(19,25)(20,28)(31,32);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31);(1,3)(2,4)(5,8)(6,7)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,29)(27,32);(1,4)(2,3)(5,6)(7,8)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8929170);