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 | |
4677 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)(25,26,29,33)(27,32,36,30)(28,37,38,31)(34,39,35,40);(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)(26,30)(28,34)(29,36)(31,39)(32,33)(35,38)(37,40);(1,2)(3,5)(4,6)(7,8)(12,21)(15,20)(18,24)(22,23)(25,28,29,38)(26,31,33,37)(27,34,36,35)(30,39,32,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24)(25,28)(26,31)(27,35)(29,38)(30,40)(32,39)(33,37)(34,36);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,26)(27,30)(28,31)(29,33)(32,36)(34,39)(35,40)(37,38);(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,27)(26,32)(28,34)(29,36)(30,33)(31,40)(35,38)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^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, 8616354);