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 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,4,5)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,32)(27,33,36,29)(28,30,38,40)(34,35,37,39);(1,3)(2,7)(4,6)(5,8)(12,19)(16,22)(18,23)(20,24)(25,27,31,36)(26,29,32,33)(28,37,38,34)(30,35,40,39);(2,8)(3,4)(10,15)(11,17)(16,22)(18,23)(26,32)(28,38)(29,33)(34,37);(1,3,6,4)(2,5,8,7)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,30)(27,34)(29,39)(31,38)(32,40)(33,35)(36,37);(1,4,6,3)(2,5,8,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,27)(28,39)(30,34)(31,33)(32,36)(35,38)(37,40);(1,5)(2,4)(3,8)(6,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,28)(27,35)(29,37)(31,40)(32,38)(33,34)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=gh,e^2=i,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, 9962997);