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 | |
4493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,31,32)(27,30,28,35)(29,33,37,40)(34,39,36,38);(1,3,6,5)(2,7,4,8)(12,19)(16,22)(18,23)(20,24)(25,27)(26,30)(28,31)(29,36)(32,35)(33,38)(34,37)(39,40);(1,3,6,5)(2,8,4,7)(10,15)(11,17)(16,22)(18,23)(26,32)(27,28)(29,37)(38,39);(1,2,6,4)(3,8,5,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,31,27)(26,30,32,35)(29,36,37,34)(33,39,40,38);(1,4,6,2)(3,8,5,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,33)(27,34)(28,36)(30,39)(31,37)(32,40)(35,38);(1,5,6,3)(2,7,4,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30)(26,27)(28,32)(29,38)(31,35)(33,36)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9944036);