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 | #53128 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 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)(10,14)(11,17)(12,20)(22,24)(25,26)(27,35)(28,39)(29,40)(30,34)(31,32)(33,36)(37,38);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(26,32)(27,36)(28,40)(34,38);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,31,36)(26,33,32,35)(28,37,40,30)(29,38,39,34);(2,4)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,29)(27,37)(30,36)(31,40)(32,39)(33,38)(34,35);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,29)(26,28)(27,38)(30,35)(31,39)(32,40)(33,37)(34,36);(1,3)(2,6)(4,7)(5,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,30,31,37)(26,34,32,38)(27,28,36,40)(29,35,39,33)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,e^2=gh,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, 8066464);