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 | |
4259 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)(9,10,15,18)(11,16,14,12)(13,17,23,24)(19,21,22,20)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,18,12)(13,19,23,22)(17,21,24,20)(27,36)(29,37)(32,35)(34,40);(1,2)(3,5)(4,6)(7,8)(9,12,15,16)(10,11,18,14)(13,21,23,20)(17,22,24,19)(25,27,31,36)(26,32,30,35)(28,29,39,37)(33,40,38,34);(1,3,4,8)(2,5,6,7)(9,12)(10,11)(13,21)(14,18)(15,16)(17,22)(19,24)(20,23)(25,28,31,39)(26,33,30,38)(27,37,36,29)(32,34,35,40);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,29,31,37)(26,34,30,40)(27,28,36,39)(32,38,35,33);(9,14,15,11)(10,12,18,16)(13,22,23,19)(17,20,24,21)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8937081);