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 | #46813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24)(25,26,32,29)(27,36,38,33)(28,34,39,30)(31,35,37,40);(1,2,4,6)(3,5,8,7)(25,27)(26,33)(28,35)(29,36)(30,37)(31,34)(32,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,28,32,39)(26,34,29,30)(27,35,38,40)(31,36,37,33);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,29,32,26)(27,33,38,36)(28,30,39,34)(31,40,37,35);(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,30)(26,28)(27,37)(29,39)(31,38)(32,34)(33,35)(36,40);(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,31,32,37)(26,35,29,40)(27,34,38,30)(28,36,39,33)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441511);