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 | |
4326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,14)(12,20)(15,17)(18,21)(25,26,31,35)(27,32,37,29)(28,33,30,34)(36,40,38,39);(1,3,6,5)(2,7,4,8)(10,14)(15,17)(16,22)(23,24)(25,27,31,37)(26,32,35,29)(28,38,30,36)(33,39,34,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,28,31,30)(26,33,35,34)(27,36,37,38)(29,39,32,40);(1,2,6,4)(3,8,5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,26)(27,29)(28,33)(30,34)(31,35)(32,37)(36,39)(38,40);(1,4,6,2)(3,8,5,7)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,20)(22,23)(25,29)(26,27)(28,39)(30,40)(31,32)(33,36)(34,38)(35,37);(1,5,6,3)(2,7,4,8)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,30,31,28)(26,34,35,33)(27,36,37,38)(29,39,32,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441591);