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 | |
4459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(9,10)(11,16)(12,17)(13,22)(14,15)(18,24)(19,23)(20,21)(25,26)(27,33)(28,39)(29,32)(30,34)(31,36)(35,38)(37,40);(2,5)(3,8)(10,15)(12,21)(16,23)(18,22)(25,27,32,40)(26,33,29,37)(28,35,34,36)(30,31,39,38);(1,3)(2,6)(4,5)(7,8)(10,15)(11,19)(17,20)(18,22)(25,28,32,34)(26,30,29,39)(27,36,40,35)(31,33,38,37);(1,4,7,6)(2,3,5,8)(9,11,14,19)(10,16,15,23)(12,22,21,18)(13,20,24,17)(25,29,32,26)(27,37,40,33)(28,30,34,39)(31,36,38,35);(1,5,7,2)(3,4,8,6)(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,16,24,23)(25,30)(26,34)(27,38)(28,29)(31,40)(32,39)(33,35)(36,37);(1,6,7,4)(2,8,5,3)(9,13,14,24)(10,18,15,22)(11,20,19,17)(12,23,21,16)(25,31,32,38)(26,35,29,36)(27,39,40,30)(28,33,34,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9080328);