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 |
| prime radical | #41122 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4330 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)(3,8)(4,5)(6,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,22)(20,24)(21,23)(25,26,31,34)(27,29,32,37)(28,30,33,39)(35,36,38,40);(1,3,6,5)(2,4,7,8)(9,11,15,18)(10,16,13,14)(12,20,23,19)(17,24,21,22)(25,27,31,32)(26,29,34,37)(28,35,33,38)(30,36,39,40);(2,7)(4,8)(10,13)(11,18)(17,21)(19,20)(27,32)(29,37)(35,38)(36,40);(2,7)(3,5)(9,12,15,23)(10,17,13,21)(11,19,18,20)(14,24,16,22)(25,28)(26,30)(27,35)(29,36)(31,33)(32,38)(34,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,10)(11,16,18,14)(12,21,23,17)(19,22,20,24)(25,29,31,37)(26,32,34,27)(28,36,33,40)(30,38,39,35);(1,5,6,3)(2,4,7,8)(9,14,15,16)(10,18,13,11)(12,22,23,24)(17,19,21,20)(25,30)(26,33)(27,36)(28,34)(29,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,d^2=gh,e^2=i,f^2=h,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, 8529472);