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 | |
4612 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)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,39,38);(3,8)(6,7)(10,15)(12,23)(16,19)(18,21)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(10,15)(11,20)(12,23)(13,24)(27,32)(28,31)(29,37)(33,40);(3,8)(6,7)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,27)(26,31)(28,34)(29,35)(30,32)(33,38)(36,40)(37,39);(1,3)(2,7)(4,6)(5,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,28,30,31)(26,32,34,27)(29,36,37,38)(33,39,40,35);(1,4,5,2)(3,6,8,7)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=i,d^2=hi,e^2=g,f^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, 8900785);