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 | |
4284 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)(3,7)(4,6)(5,8)(9,10,15,17)(11,12,13,16)(14,22,23,18)(19,24,21,20)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2,4,6)(3,5,8,7)(9,11,15,13)(10,16,17,12)(14,19,23,21)(18,24,22,20)(26,32)(27,28)(34,37)(35,36);(1,2)(3,5)(4,6)(7,8)(10,17)(12,16)(18,22)(20,24)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3,4,8)(2,5,6,7)(9,12,15,16)(10,11,17,13)(14,20,23,24)(18,19,22,21)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(9,13,15,11)(10,12,17,16)(14,21,23,19)(18,20,22,24)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,14,15,23)(10,18,17,22)(11,19,13,21)(12,20,16,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066459);