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 | #44450 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,15,17)(11,19,21,16)(12,20,23,18)(13,22,24,14)(25,26,31,33)(27,32,37,39)(28,34,35,29)(30,36,40,38);(3,7)(6,8)(9,11,15,21)(10,16,17,19)(12,22,23,14)(13,18,24,20)(25,27)(26,32)(28,38)(29,40)(30,34)(31,37)(33,39)(35,36);(1,2,5,4)(3,6,7,8)(10,17)(13,24)(16,19)(18,20)(26,33)(27,37)(29,34)(36,38);(1,3,5,7)(2,6,4,8)(9,12,15,23)(10,18,17,20)(11,14,21,22)(13,19,24,16)(25,28,31,35)(26,34,33,29)(27,36,37,38)(30,32,40,39);(1,4,5,2)(3,8,7,6)(9,13)(10,14)(11,20)(12,19)(15,24)(16,23)(17,22)(18,21)(25,29,31,34)(26,35,33,28)(27,30,37,40)(32,38,39,36);(9,14)(10,13)(11,12)(15,22)(16,18)(17,24)(19,20)(21,23)(25,30)(26,36)(27,34)(28,39)(29,37)(31,40)(32,35)(33,38)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8510770);