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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,20)(12,22)(13,19)(14,16)(15,17)(18,23)(21,24)(25,26,32,35)(27,36,39,33)(28,30,29,40)(31,34,37,38);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,16,17,20)(12,21,23,19)(13,18,24,22)(25,27)(26,33)(28,31)(29,37)(30,38)(32,39)(34,40)(35,36);(1,3)(2,5)(4,8)(6,7)(10,17)(12,23)(16,20)(19,21)(25,28,32,29)(26,30,35,40)(27,31,39,37)(33,38,36,34);(1,2)(3,5)(4,7)(6,8)(9,12)(10,18)(11,21)(13,20)(14,19)(15,23)(16,24)(17,22)(25,29,32,28)(26,30,35,40)(27,37,39,31)(33,38,36,34);(1,3)(2,6)(4,8)(5,7)(9,13,15,24)(10,19,17,21)(11,22,14,18)(12,16,23,20)(25,30)(26,29)(27,38)(28,35)(31,36)(32,40)(33,37)(34,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,11)(10,20,17,16)(12,19,23,21)(13,22,24,18)(25,31,32,37)(26,34,35,38)(27,28,39,29)(30,33,40,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=h,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, 9995597);