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 | |
4351 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)(11,17,12,14)(13,21,23,18)(19,22,20,24);(1,2,4,6)(3,5,8,7)(10,16)(11,12)(18,21)(19,20)(26,31)(27,33)(32,39)(34,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,26,30,31)(27,29,33,38)(28,32,36,39)(34,37,40,35);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,39)(36,40);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,29)(26,33)(27,31)(28,35)(30,38)(32,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,d^2=h,e^2=h,f^2=h,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, 8602498);