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 | |
4748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,16)(12,14)(13,18)(15,17)(19,24)(20,21)(22,23)(26,31)(27,34)(29,37)(36,39);(1,2)(3,5)(4,7)(6,8)(9,11)(10,16)(12,20)(13,17)(14,21)(15,18)(19,23)(22,24)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(1,2)(3,6)(4,7)(5,8)(10,17)(11,18)(12,19)(21,22)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,3)(2,5)(4,8)(6,7)(9,12,15,19)(10,14,17,24)(11,20,18,23)(13,22,16,21)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,16)(10,18,17,11)(12,22,19,21)(14,23,24,20)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(1,2)(3,5)(4,7)(6,8)(9,14)(10,19)(11,21)(12,17)(13,20)(15,24)(16,23)(18,22)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619440);