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 | #37471 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4236 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)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(1,3)(2,5)(4,8)(6,7)(10,14)(15,17)(16,20)(22,23)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(1,2)(3,5)(4,6)(7,8)(10,14)(15,17)(16,20)(22,23)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(2,6)(5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066241);