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 | |
4352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,15,19)(11,12,22,16)(13,23,21,17)(14,18,20,24)(25,26,32,29)(27,33,37,30)(28,31,34,39)(35,40,38,36);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23)(25,27,32,37)(26,33,29,30)(28,38,34,35)(31,36,39,40);(1,2)(3,4)(5,7)(6,8)(9,12,15,16)(10,11,19,22)(13,18,21,24)(14,17,20,23)(25,28,32,34)(26,31,29,39)(27,35,37,38)(30,36,33,40);(1,3,5,8)(2,6,7,4)(9,12)(10,11)(13,18)(14,23)(15,16)(17,20)(19,22)(21,24)(25,29,32,26)(27,30,37,33)(28,31,34,39)(35,40,38,36);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,20)(12,18)(14,22)(15,21)(16,24)(19,23)(25,30)(26,27)(28,36)(29,37)(31,35)(32,33)(34,40)(38,39);(1,4,5,6)(2,3,7,8)(9,14,15,20)(10,18,19,24)(11,21,22,13)(12,17,16,23)(25,31)(26,34)(27,36)(28,29)(30,38)(32,39)(33,35)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627974);