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 | |
4424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,16)(11,18,21,15)(12,20,22,17)(13,23,24,19);(3,7)(6,8)(9,11,14,21)(10,15,16,18)(12,19,22,23)(13,20,24,17)(25,26,30,34)(27,32,37,31)(28,29,33,39)(35,40,38,36);(1,2)(3,6)(4,5)(7,8)(10,16)(12,22)(15,18)(19,23)(25,27,30,37)(26,31,34,32)(28,35,33,38)(29,36,39,40);(9,12,14,22)(10,17,16,20)(11,19,21,23)(13,18,24,15)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,23)(13,17)(14,21)(15,16)(19,22)(20,24)(25,28)(26,29)(27,35)(30,33)(31,36)(32,40)(34,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,20)(12,18)(14,24)(15,22)(16,23)(17,21)(25,29)(26,33)(27,36)(28,34)(30,39)(31,38)(32,35)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9050611);