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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,31,34)(27,29,32,28)(30,38,39,33)(35,36,40,37);(1,2)(3,7)(4,8)(5,6)(10,15)(11,17)(16,22)(18,23)(25,27,31,32)(26,28,34,29)(30,35,39,40)(33,36,38,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,40,38,35);(2,6)(4,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,28)(26,27,34,32)(30,37,39,36)(33,35,38,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38);(1,4,5,7)(2,8,6,3)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658784);