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 | #43039 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(4,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(26,31)(27,34)(28,36)(37,39);(1,2,6,5)(3,8,7,4)(10,11)(15,17)(16,18)(23,24)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,3)(2,4)(5,8)(6,7)(10,11)(15,17)(16,18)(23,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(2,5)(3,7)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(1,5,6,2)(3,8,7,4)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=hi,e^2=g,f^2=h,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, 8627550);