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 |
| prime radical | #36529 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,18)(11,14,17,16)(12,21,22,13)(19,20,24,23)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(3,7)(5,8)(9,11,15,17)(10,16,18,14)(12,19,22,24)(13,20,21,23)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2)(3,5)(4,6)(7,8)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(9,12,15,22)(10,13,18,21)(11,19,17,24)(14,23,16,20)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,13)(10,12)(11,20)(14,24)(15,21)(16,19)(17,23)(18,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,14,15,16)(10,17,18,11)(12,20,22,23)(13,19,21,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131448);