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 | |
4554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26,31,34)(27,32,37,30)(28,29,40,33)(35,39,38,36);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,37)(26,32,34,30)(28,38,40,35)(29,36,33,39);(1,2)(3,5)(4,6)(7,8)(12,21)(15,20)(18,24)(22,23)(25,28,31,40)(26,33,34,29)(27,35,37,38)(30,39,32,36);(1,3,4,8)(2,5,6,7)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,28)(26,29)(27,35)(30,36)(31,40)(32,39)(33,34)(37,38);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29)(26,28)(27,36)(30,35)(31,33)(32,38)(34,40)(37,39);(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,30)(26,27)(28,39)(29,38)(31,32)(33,35)(34,37)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=g,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, 8616360);