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 | |
4458 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)(3,4)(5,6)(7,8)(9,10,14,16)(11,17,18,12)(13,15,21,22)(19,23,24,20)(25,26)(27,32)(28,34)(29,30)(31,36)(33,37)(35,40)(38,39);(2,6)(3,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,27,31,38)(26,32,36,39)(28,35,37,29)(30,34,40,33);(1,3,5,7)(2,4,6,8)(11,18)(12,17)(13,21)(15,22)(25,28,31,37)(26,33,36,34)(27,29,38,35)(30,32,40,39);(1,4)(2,7)(3,6)(5,8)(9,11,14,18)(10,12,16,17)(13,19,21,24)(15,20,22,23)(25,28)(26,34)(27,35)(29,38)(30,39)(31,37)(32,40)(33,36);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29,31,35)(26,30,36,40)(27,37,38,28)(32,33,39,34);(9,13)(10,15)(11,19)(12,20)(14,21)(16,22)(17,23)(18,24)(25,30)(26,35)(27,34)(28,39)(29,36)(31,40)(32,37)(33,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=ghi,e^2=h,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, 8900137);