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 | #36136 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(26,31)(27,32)(29,37)(35,39);(1,2,5,8)(3,4,7,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,2)(3,6)(4,7)(5,8)(25,27,30,32)(26,28,31,36)(29,34,37,40)(33,35,38,39);(3,7)(4,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26)(27,28)(29,38)(30,31)(32,36)(33,37)(34,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,21)(13,19)(15,24)(16,17)(18,22)(20,23)(25,28)(26,32)(27,31)(29,39)(30,36)(33,34)(35,37)(38,40);(1,4,5,6)(2,7,8,3)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,29,30,37)(26,33,31,38)(27,34,32,40)(28,35,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9129370);