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 | #38948 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 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)(11,18)(12,15)(13,16)(14,20)(17,22)(19,23)(21,24)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,2,4,6)(3,5,8,7)(11,13)(16,18)(19,21)(23,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,39)(33,36)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,28)(26,31)(27,34)(29,36)(30,38)(32,40)(33,35)(37,39);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,27)(26,30)(28,33)(29,35)(31,37)(32,39)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620544);