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 | |
4500 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)(4,5)(6,8)(9,10)(11,18)(12,17)(13,22)(14,15)(16,21)(19,24)(20,23)(25,26,32,35)(27,36,38,33)(28,30,29,39)(31,34,40,37);(2,5)(6,7)(10,15)(12,22)(16,18)(19,23)(25,27)(26,33)(28,34)(29,37)(30,31)(32,38)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,17)(16,18)(20,24)(25,28,32,29)(26,30,35,39)(27,34,38,37)(31,36,40,33);(1,3,4,8)(2,6,5,7)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,29,32,28)(26,30,35,39)(27,37,38,34)(31,36,40,33);(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,30)(26,29)(27,31)(28,35)(32,39)(33,37)(34,36)(38,40);(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,31)(26,34)(27,30)(28,33)(29,36)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8806084);