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 | #43535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4247 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)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,31,29)(27,35,37,32)(28,38,39,33)(30,40,36,34);(1,3,5,4)(2,6,7,8)(10,14)(15,17)(16,20)(22,23)(25,27,31,37)(26,32,29,35)(28,36,39,30)(33,34,38,40);(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,31,39)(26,33,29,38)(27,30,37,36)(32,40,35,34);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,34)(35,40);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,29,31,26)(27,32,37,35)(28,38,39,33)(30,40,36,34);(1,2)(3,6)(4,8)(5,7)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30,31,36)(26,34,29,40)(27,28,37,39)(32,38,35,33)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9613525);