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 | |
4607 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)(4,5)(6,8)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26,31,34)(27,32,30,28)(29,33,39,40)(35,37,38,36);(2,5)(6,7)(10,15)(12,23)(16,19)(18,21)(25,27,31,30)(26,32,34,28)(29,35,39,38)(33,37,40,36);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(12,23)(13,24)(25,28,31,32)(26,27,34,30)(29,37,39,36)(33,38,40,35);(1,3)(2,6)(4,8)(5,7)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,28)(26,27)(29,37)(30,34)(31,32)(33,38)(35,40)(36,39);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40);(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30,31,27)(26,28,34,32)(29,38,39,35)(33,36,40,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=h,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, 8937331);