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 | |
4480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,13,19,15)(12,17,21,24)(18,22,20,23)(25,26,31,34)(27,32,36,30)(28,37,29,33)(35,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,21)(23,24)(25,27)(26,32)(28,38)(29,35)(30,34)(31,36)(33,40)(37,39);(1,2,4,6)(3,7,8,5)(10,16)(12,21)(13,15)(18,20)(25,28,31,29)(26,33,34,37)(27,35,36,38)(30,40,32,39);(3,8)(5,7)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,21)(23,24)(25,29,31,28)(26,33,34,37)(27,35,36,38)(30,39,32,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,18)(13,22)(14,21)(15,23)(16,24)(19,20)(25,30)(26,27)(28,39)(29,40)(31,32)(33,38)(34,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,13,14,15)(10,11,16,19)(12,22,21,23)(17,18,24,20)(25,27)(26,30)(28,38)(29,35)(31,36)(32,34)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8850165);