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 | |
4465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,16)(11,14,12,17)(13,21,23,18)(19,22,20,24)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,15)(13,19)(16,17)(18,24)(20,23)(21,22)(26,32)(27,28)(34,37)(35,36);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(18,21)(19,20)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,22,21,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,24,17,22)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,14,15,17)(10,12,16,11)(13,22,23,24)(18,19,21,20)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627711);