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 | |
4538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,19,20,16)(12,22,23,14)(13,24,21,18)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,17,19)(12,21,23,13)(14,18,22,24)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(1,2,5,4)(3,6,8,7)(10,17)(11,20)(12,23)(18,24)(26,33)(28,34)(32,36)(35,40);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,21)(13,20)(15,23)(16,18)(17,22)(19,24)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(9,13,15,21)(10,18,17,24)(11,12,20,23)(14,16,22,19)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(9,14)(10,12)(11,18)(13,19)(15,22)(16,21)(17,23)(20,24)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7808589);