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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,13,14)(12,18,22,24)(19,21,20,23)(25,26,32,37)(27,30,39,33)(28,35,29,34)(31,36,38,40);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,20)(13,15)(14,17)(18,23)(19,22)(21,24)(25,27)(26,33)(28,38)(29,31)(30,37)(32,39)(34,40)(35,36);(2,5)(6,8)(10,17)(11,13)(18,24)(19,20)(25,28,32,29)(26,34,37,35)(27,38,39,31)(30,36,33,40);(3,7)(6,8)(9,12,15,22)(10,18,17,24)(11,19,13,20)(14,23,16,21)(25,29,32,28)(26,35,37,34)(27,38,39,31)(30,36,33,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,16,17,14)(12,20,22,19)(18,21,24,23)(25,30)(26,27)(28,40)(29,36)(31,34)(32,33)(35,38)(37,39);(1,3,4,7)(2,6,5,8)(9,14,15,16)(10,11,17,13)(12,21,22,23)(18,20,24,19)(25,31,32,38)(26,36,37,40)(27,29,39,28)(30,34,33,35)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8543476);