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 | |
4327 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)(3,5)(4,7)(6,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(25,26,31,32)(27,30,33,40)(28,29,37,38)(34,39,36,35);(2,3)(6,7)(10,11)(15,17)(16,18)(23,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,2,5,3)(4,6,8,7)(10,11)(15,17)(16,18)(23,24)(26,32)(27,33)(28,37)(35,39);(1,3,5,2)(4,7,8,6)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,28)(26,29)(27,34)(30,39)(31,37)(32,38)(33,36)(35,40);(1,4)(2,6)(3,7)(5,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29)(26,28)(27,35)(30,36)(31,38)(32,37)(33,39)(34,40);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30,31,40)(26,33,32,27)(28,35,37,39)(29,34,38,36)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=g,e^2=hi,f^2=gi,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, 8633674);