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 | #2669 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(25,26)(27,31)(28,35)(29,34)(30,33)(32,36)(37,39)(38,40);(1,3,6,5)(2,7,8,4)(9,10,14,15)(11,16,18,12)(13,17,23,24)(19,22,21,20)(25,27,32,38)(26,31,36,40)(28,29,30,37)(33,39,35,34);(1,3,6,5)(2,7,8,4)(10,15)(11,18)(17,24)(19,21)(25,28,32,30)(26,33,36,35)(27,37,38,29)(31,34,40,39);(2,8)(3,5)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,29,32,37)(26,34,36,39)(27,28,38,30)(31,35,40,33);(1,4)(2,3)(5,8)(6,7)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,30,32,28)(26,35,36,33)(27,37,38,29)(31,34,40,39);(1,5,6,3)(2,7,8,4)(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,31,32,40)(26,27,36,38)(28,34,30,39)(29,33,37,35)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633461);