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 | #40054 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(4,8)(25,26,31,36)(27,37,39,32)(28,33,38,34)(29,35,40,30);(1,2,6,5)(3,8,7,4)(9,10,14,15)(11,16,18,12)(13,17,23,24)(19,22,21,20)(25,27)(26,32)(28,35)(29,33)(30,38)(31,39)(34,40)(36,37);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(17,24)(19,21)(25,28,31,38)(26,33,36,34)(27,35,39,30)(29,37,40,32);(2,5)(3,7)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,28)(26,34)(27,35)(29,37)(30,39)(31,38)(32,40)(33,36);(1,4)(2,3)(5,7)(6,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,29,31,40)(26,35,36,30)(27,33,39,34)(28,37,38,32);(1,5,6,2)(3,8,7,4)(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,30,31,35)(26,29,36,40)(27,38,39,28)(32,33,37,34)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,e^2=ghi,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, 8633497);