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 | |
4522 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)(4,5)(6,8)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(25,26,29,32)(27,33,34,30)(28,31,38,37)(35,40,36,39);(2,5)(6,7)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(27,34)(30,33)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,30)(28,35)(29,34)(31,39)(32,33)(36,38)(37,40);(1,3,4,8)(2,6,5,7)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18)(25,27)(26,30)(28,36)(29,34)(31,40)(32,33)(35,38)(37,39);(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,28,29,38)(26,31,32,37)(27,35,34,36)(30,39,33,40);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,26)(27,30)(28,37)(29,32)(31,38)(33,34)(35,40)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10371727);