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 | #37469 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,18)(12,15)(13,20)(14,16)(17,22)(19,23)(21,24)(25,26,30,32)(27,33,36,31)(28,37,38,29)(34,35,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,26,30,32)(27,31,36,33)(28,37,38,29)(34,39,40,35);(1,2,4,5)(3,6,7,8)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,28,30,38)(26,29,32,37)(27,34,36,40)(31,35,33,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29)(26,28)(27,35)(30,37)(31,34)(32,38)(33,40)(36,39);(1,3,4,7)(2,6,5,8)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,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, 8066126);