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 | #54948 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,22,16)(12,14,23,18)(13,19,24,21)(25,26)(27,29)(28,36)(30,34)(31,32)(33,39)(35,38)(37,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,22)(10,16,17,20)(12,13,23,24)(14,21,18,19)(26,32)(27,34)(33,36)(35,40);(2,6)(3,7)(10,17)(12,23)(13,24)(16,20)(25,27,31,34)(26,29,32,30)(28,35,39,40)(33,37,36,38);(3,7)(4,8)(9,12,15,23)(10,18,17,14)(11,13,22,24)(16,19,20,21)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(1,3)(2,4)(5,7)(6,8)(9,13)(10,19)(11,12)(14,20)(15,24)(16,18)(17,21)(22,23)(25,29,31,30)(26,27,32,34)(28,37,39,38)(33,35,36,40);(1,4)(2,3)(5,8)(6,7)(9,14,15,18)(10,12,17,23)(11,21,22,19)(13,20,24,16)(25,30,31,29)(26,34,32,27)(28,38,39,37)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532666);