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 | |
4728 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)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(26,30)(31,37)(32,35)(38,40);(2,6)(3,7)(10,15)(11,20)(17,22)(18,21)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,3,5,7)(2,4,6,8)(10,15)(12,23)(16,19)(18,21)(27,33)(31,37)(34,39)(38,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,23)(22,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441957);