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 | #48715 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(12,20)(15,22)(18,24)(21,23)(25,26)(27,33)(28,39)(29,32)(30,34)(31,37)(35,40)(36,38);(1,3,6,4)(2,7,5,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,27,32,38)(26,33,29,36)(28,37,30,35)(31,34,40,39);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,32,30)(26,34,29,39)(27,35,38,37)(31,36,40,33);(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,29,32,26)(27,36,38,33)(28,34,30,39)(31,37,40,35);(1,4,6,3)(2,7,5,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,30,32,28)(26,34,29,39)(27,37,38,35)(31,36,40,33);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,31,32,40)(26,35,29,37)(27,34,38,39)(28,36,30,33)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620052);