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 | |
4473 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)(11,18)(12,21)(13,15)(14,17)(16,22)(19,23)(20,24)(25,26,31,32)(27,28,29,33)(30,38,39,34)(35,40,37,36);(1,3)(2,7)(4,5)(6,8)(9,11,14,13)(10,15,17,18)(12,19,22,20)(16,23,21,24)(26,32)(27,29)(34,38)(35,37);(1,2,6,4)(3,7,8,5)(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,21)(18,24)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(2,4)(3,8)(9,12)(10,16)(11,20)(13,19)(14,22)(15,24)(17,21)(18,23)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(1,4,6,2)(3,7,8,5)(9,13,14,11)(10,15,17,18)(12,20,22,19)(16,23,21,24)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(1,5,6,7)(2,3,4,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,24)(21,22)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,38)(33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=i,e^2=g,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, 10227583);