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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26,31,35)(27,29,36,32)(28,33,37,38)(30,34,39,40);(2,5)(6,8)(10,15)(11,19)(12,22)(21,23)(27,36)(28,37)(29,32)(33,38);(1,2,4,5)(3,6,7,8)(10,15)(12,22)(16,18)(20,24)(25,27,31,36)(26,32,35,29)(28,30,37,39)(33,40,38,34);(3,7)(6,8)(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,23,17,21)(25,28)(26,33)(27,30)(29,34)(31,37)(32,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,29,31,32)(26,27,35,36)(28,34,37,40)(30,38,39,33);(1,3)(2,6)(4,7)(5,8)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,30)(26,34)(27,37)(28,36)(29,38)(31,39)(32,33)(35,40)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506728);