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 | |
4403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,15)(11,18,16,12)(13,17,23,24)(19,22,21,20)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,39)(38,40);(2,5)(6,7)(10,15)(11,16)(17,24)(19,21)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,24)(19,21)(26,31)(27,32)(33,37)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,18)(13,21,23,19)(17,22,24,20)(25,26)(27,34)(28,32)(29,37)(30,31)(33,39)(35,36)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,28,30,34)(26,32,31,27)(29,38,39,36)(33,35,37,40);(9,13,14,23)(10,17,15,24)(11,19,16,21)(12,20,18,22)(25,29,30,39)(26,33,31,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8631974);