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 | #35715 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4370 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)(3,7,8,5)(9,10,15,19)(11,16,14,18)(12,17,23,13)(20,22,21,24)(25,26,31,34)(27,29,36,33)(28,37,30,32)(35,39,38,40);(1,3,4,8)(2,5,6,7)(9,11,15,14)(10,16,19,18)(12,21,23,20)(13,22,17,24)(25,27)(26,29)(28,38)(30,35)(31,36)(32,39)(33,34)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,17,19,13)(11,20,14,21)(16,22,18,24)(25,28,31,30)(26,32,34,37)(27,35,36,38)(29,40,33,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,21)(14,20)(15,23)(16,22)(17,19)(18,24)(25,29,31,33)(26,27,34,36)(28,39,30,40)(32,38,37,35);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,22)(14,24)(15,17)(16,21)(18,20)(19,23)(25,27)(26,33)(28,35)(29,34)(30,38)(31,36)(32,39)(37,40);(9,14,15,11)(10,18,19,16)(12,21,23,20)(13,22,17,24)(25,30,31,28)(26,32,34,37)(27,35,36,38)(29,39,33,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657133);