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 | #45514 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,22)(11,23,24,17)(12,19,13,18)(14,15,21,20)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,5,4)(3,6,7,8)(9,11)(10,17)(12,21)(13,14)(15,19)(16,24)(18,20)(22,23)(28,36)(31,38)(34,40)(37,39);(2,4)(6,8)(9,12,16,13)(10,18,22,19)(11,21,24,14)(15,17,20,23)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(3,7)(6,8)(9,13,16,12)(10,19,22,18)(11,21,24,14)(15,17,20,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,14,16,21)(10,20,22,15)(11,13,24,12)(17,18,23,19)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,4,5,2)(3,8,7,6)(9,15,16,20)(10,21,22,14)(11,19,24,18)(12,23,13,17)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8232006);