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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4278 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)(9,10)(11,16)(12,15)(13,22)(14,23)(17,20)(18,24)(19,21)(25,26,31,37)(27,35,29,32)(28,34,39,33)(30,38,40,36);(3,8)(5,6)(9,11,15,19)(10,16,12,21)(13,14,20,24)(17,18,22,23)(25,27,31,29)(26,32,37,35)(28,38,39,36)(30,33,40,34);(1,3)(2,5)(4,8)(6,7)(10,12)(11,19)(14,24)(17,22)(25,28,31,39)(26,33,37,34)(27,38,29,36)(30,32,40,35);(1,3,4,8)(2,6,7,5)(9,12,15,10)(11,16,19,21)(13,17,20,22)(14,23,24,18)(25,28)(26,34)(27,36)(29,38)(30,35)(31,39)(32,40)(33,37);(1,3)(2,6)(4,8)(5,7)(9,13,15,20)(10,17,12,22)(11,14,19,24)(16,18,21,23)(25,29,31,27)(26,35,37,32)(28,36,39,38)(30,34,40,33);(1,2)(3,6)(4,7)(5,8)(9,14)(10,18)(11,20)(12,23)(13,19)(15,24)(16,22)(17,21)(25,30,31,40)(26,36,37,38)(27,33,29,34)(28,35,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841855);