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 | |
4211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18)(26,32)(28,33)(34,39)(36,40);(1,2)(3,8)(4,6)(5,7)(10,15)(12,22)(16,19)(18,23)(25,26,31,32)(27,33,30,28)(29,34,37,39)(35,40,38,36);(1,3,5,4)(2,6,7,8)(10,15)(11,20)(17,21)(18,23)(25,27,31,30)(26,33,32,28)(29,35,37,38)(34,40,39,36);(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,24)(22,23)(25,28)(26,27)(29,36)(30,32)(31,33)(34,35)(37,40)(38,39);(1,4,5,3)(2,6,7,8)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,16,24,19)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,30,31,27)(26,28,32,33)(29,35,37,38)(34,40,39,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9614080);