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)(3,7)(4,5)(6,8)(25,26,32,36)(27,37,39,33)(28,30,29,34)(31,35,38,40);(2,5)(6,7)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,27)(26,33)(28,35)(29,40)(30,31)(32,39)(34,38)(36,37);(1,2)(3,6)(4,5)(7,8)(10,15)(13,22)(16,19)(20,24)(25,28,32,29)(26,34,36,30)(27,35,39,40)(31,33,38,37);(1,3,4,8)(2,6,5,7)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,29,32,28)(26,30,36,34)(27,35,39,40)(31,33,38,37);(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,30)(26,28)(27,38)(29,36)(31,39)(32,34)(33,40)(35,37);(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,31,32,38)(26,35,36,40)(27,30,39,34)(28,37,29,33)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8137859);