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 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,14)(11,17,18,15)(12,16,21,22)(19,24,20,23)(25,26,32,34)(27,28,30,31)(29,33,38,40)(35,36,37,39);(2,3)(6,7)(10,14)(11,18)(16,22)(19,20)(25,27)(26,28)(29,37)(30,32)(31,34)(33,39)(35,38)(36,40);(1,2,5,3)(4,6,8,7)(10,14)(15,17)(16,22)(23,24)(25,28,32,31)(26,27,34,30)(29,36,38,39)(33,35,40,37);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,20)(13,18)(14,17)(16,24)(19,21)(22,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,39)(32,38)(34,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,30,32,27)(26,28,34,31)(29,37,38,35)(33,36,40,39);(9,12,13,21)(10,16,14,22)(11,19,18,20)(15,23,17,24)(25,31,32,28)(26,27,34,30)(29,36,38,39)(33,37,40,35)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8556124);