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 | |
4619 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)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,32,35)(27,36,38,33)(28,30,29,39)(31,34,37,40);(1,2,4,6)(3,5,8,7)(12,19)(15,17)(20,24)(22,23)(25,27,32,38)(26,33,35,36)(28,37,29,31)(30,34,39,40);(1,2)(3,5)(4,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,28,32,29)(26,30,35,39)(27,37,38,31)(33,34,36,40);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,29,32,28)(26,30,35,39)(27,31,38,37)(33,34,36,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,30)(26,29)(27,34)(28,35)(31,33)(32,39)(36,37)(38,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,31)(26,34)(27,28)(29,38)(30,36)(32,37)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10013812);