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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,14)(11,17)(16,22)(18,19)(25,26,31,35)(27,36,28,32)(29,33,39,37)(30,34,40,38);(3,7)(6,8)(9,10,13,14)(11,15,17,21)(12,16,20,22)(18,23,19,24)(27,28)(29,39)(32,36)(33,37);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,32,35,36)(29,34,39,38)(30,37,40,33);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,17)(14,21)(16,24)(18,20)(22,23)(25,28,31,27)(26,32,35,36)(29,38,39,34)(30,37,40,33);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,29,31,39)(26,33,35,37)(27,34,28,38)(30,36,40,32);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,30)(26,34)(27,37)(28,33)(29,36)(31,40)(32,39)(35,38)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400041);