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 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,3,6,5)(2,7,4,8)(10,11)(15,17)(16,18)(23,24)(27,34)(30,31)(35,40)(38,39);(3,5)(7,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,4,6,2)(3,8,5,7)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(1,5,6,3)(2,7,4,8)(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635423);