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 | |
4629 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,32)(28,30)(29,34)(31,33)(35,37)(36,39)(38,40);(1,3,7,6)(2,4,8,5)(11,17)(15,21)(18,23)(22,24)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,39,40,37);(1,4,7,5)(2,3,8,6)(10,14)(15,21)(16,19)(22,24)(26,33)(27,28)(34,40)(35,36);(2,8)(4,5)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,27)(26,30,33,32)(29,35,38,36)(34,37,40,39);(1,5,7,4)(2,3,8,6)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,38)(26,34,33,40)(27,35,28,36)(30,39,32,37);(1,6,7,3)(2,5,8,4)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,27)(28,33)(29,37)(31,32)(34,36)(35,40)(38,39)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10078570);