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 | #2414 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4276 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)(25,26,32,37)(27,33,39,40)(28,36,30,34)(29,35,38,31);(2,3)(6,7)(9,10,14,15)(11,16,18,12)(13,17,23,24)(19,22,21,20)(25,27,32,39)(26,33,37,40)(28,29,30,38)(31,36,35,34);(1,2,5,3)(4,6,8,7)(10,15)(11,18)(17,24)(19,21)(25,28,32,30)(26,34,37,36)(27,38,39,29)(31,33,35,40);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,29,32,38)(26,35,37,31)(27,28,39,30)(33,36,40,34);(1,4,5,8)(2,6,3,7)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,30,32,28)(26,36,37,34)(27,38,39,29)(31,33,35,40);(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,31,32,35)(26,29,37,38)(27,36,39,34)(28,33,30,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=h,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, 8633469);