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:
(1,2)(3,7)(4,6)(5,8)(10,14)(15,17)(16,21)(22,23)(25,26,31,33)(27,34,29,32)(28,37,39,30)(35,40,38,36);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,31,29)(26,32,33,34)(28,38,39,35)(30,36,37,40);(1,2)(3,5)(4,6)(7,8)(12,20)(16,21)(18,24)(22,23)(26,33)(30,37)(32,34)(36,40);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,30)(27,35)(29,38)(31,39)(32,40)(33,37)(34,36);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,27)(26,34,33,32)(28,35,39,38)(30,40,37,36);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,38)(33,39)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202827);