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 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,17)(12,13)(14,22)(15,16)(18,23)(19,20)(21,24)(25,26,31,32)(27,28,35,40)(29,33,30,34)(36,38,37,39);(1,3,6,5)(2,7,4,8)(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,22,24)(26,32)(27,35)(33,34)(36,37);(1,2,6,4)(3,7,5,8)(10,16)(12,17)(18,22)(20,24)(25,27)(26,28)(29,36)(30,37)(31,35)(32,40)(33,38)(34,39);(2,4)(7,8)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,28)(26,27)(29,38)(30,39)(31,40)(32,35)(33,36)(34,37);(1,4,6,2)(3,8,5,7)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,29,31,30)(26,33,32,34)(27,36,35,37)(28,38,40,39);(1,5,6,3)(2,8,4,7)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,30,31,29)(26,34,32,33)(27,37,35,36)(28,39,40,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,e^2=g,f^2=gh,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, 8066168);