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 | |
4608 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)(9,10,14,16)(11,18,21,15)(12,20,22,17)(13,23,24,19)(25,26)(27,33)(28,31)(29,34)(30,36)(32,37)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,16,18)(12,19,22,23)(13,20,24,17)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40);(1,2)(3,5)(4,6)(7,8)(10,16)(12,22)(15,18)(19,23)(25,28,29,37)(26,31,34,32)(27,35,36,40)(30,38,33,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,22)(10,17,16,20)(11,19,21,23)(13,18,24,15)(25,28)(26,32)(27,35)(29,37)(30,39)(31,34)(33,38)(36,40);(9,11)(10,18)(12,23)(13,17)(14,21)(15,16)(19,22)(20,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,38)(39,40);(9,13)(10,19)(11,20)(12,18)(14,24)(15,22)(16,23)(17,21)(25,27)(26,33)(28,35)(29,36)(30,34)(31,39)(32,38)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9050545);