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 | |
4698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(9,10)(11,13)(12,17)(14,16)(15,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,38)(29,33)(30,39)(31,34)(35,36)(37,40);(2,5)(3,8)(9,11,14,18)(10,15,16,13)(12,19,20,24)(17,23,21,22)(25,27,31,37)(26,32,34,40)(28,33,39,35)(29,30,36,38);(1,3)(2,6)(4,5)(7,8)(10,16)(12,20)(13,15)(19,24)(25,28,31,39)(26,30,34,38)(27,35,37,33)(29,40,36,32);(1,4,7,6)(2,3,5,8)(9,12,14,20)(10,17,16,21)(11,19,18,24)(13,22,15,23)(25,28)(26,30)(27,33)(29,40)(31,39)(32,36)(34,38)(35,37);(1,5,7,2)(3,4,8,6)(9,10)(11,15)(12,21)(13,18)(14,16)(17,20)(19,22)(23,24)(25,29,31,36)(26,33,34,35)(27,30,37,38)(28,32,39,40);(1,6,7,4)(2,8,5,3)(9,13,14,15)(10,18,16,11)(12,22,20,23)(17,24,21,19)(25,30)(26,28)(27,36)(29,37)(31,38)(32,33)(34,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961688);