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 | |
4279 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)(9,10,15,18)(11,16,14,24)(12,17,22,13)(19,20,21,23)(25,26,31,29)(27,32,35,28)(30,37,39,33)(34,36,38,40);(1,3)(2,7)(4,5)(6,8)(9,11,15,14)(10,16,18,24)(12,19,22,21)(13,23,17,20)(25,27,31,35)(26,32,29,28)(30,34,39,38)(33,40,37,36);(1,2,6,4)(3,7,8,5)(9,12,15,22)(10,13,18,17)(11,19,14,21)(16,23,24,20)(25,28,31,32)(26,27,29,35)(30,36,39,40)(33,34,37,38);(2,4)(3,8)(9,12)(10,17)(11,19)(13,18)(14,21)(15,22)(16,20)(23,24)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,40,38,36);(1,4,6,2)(3,7,8,5)(9,13,15,17)(10,12,18,22)(11,20,14,23)(16,21,24,19)(25,27)(26,32)(28,29)(30,38)(31,35)(33,36)(34,39)(37,40);(1,5,6,7)(2,3,4,8)(9,14,15,11)(10,16,18,24)(12,21,22,19)(13,23,17,20)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,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, 10227617);