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 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,16,19)(11,20,14,17)(12,15,13,18)(21,22,23,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,37)(35,40)(36,39);(1,3,6,5)(2,7,4,8)(9,11,16,14)(10,17,19,20)(12,23,13,21)(15,22,18,24)(27,35)(28,37)(31,40)(32,38);(9,12,16,13)(10,18,19,15)(11,21,14,23)(17,22,20,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,33)(34,40)(38,39);(1,2,6,4)(3,8,5,7)(9,13,16,12)(10,15,19,18)(11,21,14,23)(17,22,20,24)(25,28,30,37)(26,32,34,38)(27,36,35,29)(31,33,40,39);(1,4,6,2)(3,8,5,7)(9,14,16,11)(10,17,19,20)(12,23,13,21)(15,24,18,22)(25,29,30,36)(26,33,34,39)(27,37,35,28)(31,32,40,38);(1,5,6,3)(2,7,4,8)(9,15)(10,12)(11,22)(13,19)(14,24)(16,18)(17,23)(20,21)(25,26)(27,31)(28,32)(29,39)(30,34)(33,36)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10426564);