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 | |
4460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(12,20)(16,21)(18,24)(22,23)(25,26)(27,34)(28,31)(29,33)(30,36)(32,37)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,21)(22,23)(25,28,29,37)(26,31,33,32)(27,35,36,40)(30,38,34,39);(2,5)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,32)(27,35)(29,37)(30,39)(31,33)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,26)(27,30)(28,31)(29,33)(32,37)(34,36)(35,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=i,f^2=i,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, 8081849);