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 | #50013 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(25,26)(27,36)(28,33)(29,38)(30,34)(31,35)(32,39)(37,40);(3,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,24)(20,23)(21,22)(25,27,31,39)(26,32,35,36)(28,37,29,34)(30,38,40,33);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,22)(21,24)(25,28,31,29)(26,33,35,38)(27,37,39,34)(30,36,40,32);(9,11)(10,15)(12,20)(13,19)(14,17)(16,23)(18,22)(21,24)(25,29,31,28)(26,33,35,38)(27,34,39,37)(30,36,40,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,21)(13,17)(15,19)(16,22)(18,24)(20,23)(25,26)(27,32)(28,33)(29,38)(30,37)(31,35)(34,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,21)(19,20)(25,30,31,40)(26,34,35,37)(27,38,39,33)(28,36,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9117707);