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 | |
4589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,21)(19,24)(25,26,31,35)(27,30,36,32)(28,37,39,33)(29,34,40,38);(1,2)(3,8)(4,5)(6,7)(9,10,14,15)(11,16,18,12)(13,21,22,17)(19,20,24,23)(25,27,31,36)(26,32,35,30)(28,29,39,40)(33,34,37,38);(2,7)(3,4)(11,18)(12,16)(19,24)(20,23)(25,28,31,39)(26,33,35,37)(27,29,36,40)(30,38,32,34);(1,3,6,4)(2,8,7,5)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24)(25,28)(26,33)(27,29)(30,38)(31,39)(32,34)(35,37)(36,40);(1,4,6,3)(2,5,7,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29)(26,34)(27,28)(30,37)(31,40)(32,33)(35,38)(36,39);(1,5)(2,3)(4,7)(6,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30,31,32)(26,27,35,36)(28,38,39,34)(29,37,40,33)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8705728);