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 | |
4586 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)(4,8)(6,7)(9,10,15,17)(11,16,13,19)(12,18,23,22)(14,21,24,20)(25,26,30,32)(27,35,36,31)(28,37,38,33)(29,34,39,40);(2,7)(3,8)(9,11)(10,16)(12,21)(13,15)(14,22)(17,19)(18,24)(20,23)(25,27,30,36)(26,31,32,35)(28,29,38,39)(33,34,37,40);(1,3,6,8)(2,4,7,5)(10,17)(11,13)(12,23)(14,24)(26,32)(28,38)(29,39)(31,35);(1,4,6,5)(2,3,7,8)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,28,30,38)(26,33,32,37)(27,29,36,39)(31,34,35,40);(1,2)(3,4)(5,8)(6,7)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,29)(26,34)(27,28)(30,39)(31,33)(32,40)(35,37)(36,38);(1,5,6,4)(2,8,7,3)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8937249);