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 | |
4457 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)(3,5,8,7)(9,10,15,17)(11,20,21,16)(12,22,19,14)(13,18,23,24)(25,26,31,33)(27,32,30,40)(28,34,37,29)(35,36,38,39);(1,3)(2,7)(4,5)(6,8)(9,11)(10,16)(12,13)(14,18)(15,21)(17,20)(19,23)(22,24)(25,27,31,30)(26,32,33,40)(28,35,37,38)(29,39,34,36);(1,2,6,4)(3,7,8,5)(10,17)(11,21)(12,19)(18,24)(26,33)(28,37)(32,40)(35,38);(2,4)(3,8)(9,12,15,19)(10,14,17,22)(11,13,21,23)(16,18,20,24)(25,28)(26,34)(27,35)(29,33)(30,38)(31,37)(32,36)(39,40);(1,4,6,2)(3,7,8,5)(9,13)(10,18)(11,19)(12,21)(14,20)(15,23)(16,22)(17,24)(25,29,31,34)(26,28,33,37)(27,36,30,39)(32,38,40,35);(1,5,6,7)(2,3,4,8)(9,14)(10,19)(11,18)(12,17)(13,20)(15,22)(16,23)(21,24)(25,30,31,27)(26,32,33,40)(28,38,37,35)(29,39,34,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227610);