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 | #37043 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(25,26)(27,38)(28,34)(29,39)(30,35)(31,36)(32,37)(33,40);(3,7)(6,8)(10,16)(11,12)(18,21)(19,20)(25,27)(26,33)(28,35)(29,31)(30,39)(32,40)(34,36)(37,38);(1,2,5,4)(3,6,7,8)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,28,32,29)(26,34,37,39)(27,35,40,31)(30,33,36,38);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,29,32,28)(26,34,37,39)(27,35,40,31)(30,38,36,33);(1,4,5,2)(3,8,7,6)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,30)(26,35)(27,39)(28,33)(29,38)(31,37)(32,36)(34,40);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,31,32,35)(26,36,37,30)(27,29,40,28)(33,34,38,39)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=g,e^2=h,f^2=g,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, 8603306);