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 | #49563 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,16,14)(11,17,22,21)(12,23,13,18)(15,24,20,19)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11,16,22)(10,17,14,21)(12,20,13,15)(18,24,23,19)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(9,12,16,13)(10,18,14,23)(11,20,22,15)(17,24,21,19)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,13,16,12)(10,18,14,23)(11,15,22,20)(17,24,21,19)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,4,5,3)(2,6,7,8)(9,14,16,10)(11,21,22,17)(12,23,13,18)(15,24,20,19)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,6)(4,8)(5,7)(9,15)(10,19)(11,12)(13,22)(14,24)(16,20)(17,18)(21,23)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441506);