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 | #41821 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,18)(12,21)(13,16)(14,22)(15,17)(19,20)(23,24)(25,26,31,32)(27,30,34,29)(28,36,39,33)(35,37,40,38);(1,3,5,8)(2,6,4,7)(26,32)(27,34)(33,36)(35,40);(1,2,5,4)(3,6,8,7)(9,11,15,13)(10,16,17,18)(12,19,22,23)(14,20,21,24)(25,27,31,34)(26,29,32,30)(28,35,39,40)(33,37,36,38);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,19)(13,23)(15,22)(16,20)(17,21)(18,24)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(9,13,15,11)(10,16,17,18)(12,19,22,23)(14,24,21,20)(25,29,31,30)(26,27,32,34)(28,37,39,38)(33,35,36,40);(1,4,5,2)(3,7,8,6)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,30,31,29)(26,34,32,27)(28,38,39,37)(33,40,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553021);