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 | #47681 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,19,23)(12,14,21,24)(13,18,20,22)(25,26,31,33)(27,34,30,32)(28,37,38,29)(35,40,39,36);(1,2)(3,6)(4,7)(5,8)(9,11,15,19)(10,16,17,23)(12,20,21,13)(14,22,24,18)(25,27,31,30)(26,32,33,34)(28,35,38,39)(29,40,37,36);(1,3)(2,5)(4,8)(6,7)(10,17)(11,19)(12,21)(18,22)(26,33)(28,38)(32,34)(35,39);(1,2)(3,5)(4,7)(6,8)(9,12)(10,14)(11,20)(13,19)(15,21)(16,22)(17,24)(18,23)(25,28,31,38)(26,29,33,37)(27,35,30,39)(32,40,34,36);(1,3)(2,6)(4,8)(5,7)(9,13,15,20)(10,18,17,22)(11,21,19,12)(14,16,24,23)(25,29)(26,28)(27,36)(30,40)(31,37)(32,39)(33,38)(34,35);(1,2)(3,5)(4,7)(6,8)(9,14)(10,12)(11,22)(13,23)(15,24)(16,20)(17,21)(18,19)(25,30,31,27)(26,34,33,32)(28,39,38,35)(29,36,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9994840);