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 | |
4559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,12)(11,13,16,19)(14,20,23,17)(18,22,24,21)(25,26,31,30)(27,35,37,32)(28,33,39,38)(29,40,36,34);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,19)(14,18)(15,16)(17,21)(20,22)(23,24)(25,27)(26,32)(28,36)(29,39)(30,35)(31,37)(33,40)(34,38);(1,3,5,6)(2,4,7,8)(10,12)(11,16)(17,20)(18,24)(25,28,31,39)(26,33,30,38)(27,36,37,29)(32,40,35,34);(1,2,5,7)(3,4,6,8)(9,12,15,10)(11,13,16,19)(14,20,23,17)(18,21,24,22)(25,29)(26,34)(27,28)(30,40)(31,36)(32,33)(35,38)(37,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,16)(11,12)(14,22)(15,19)(17,18)(20,24)(21,23)(25,27)(26,35)(28,36)(29,39)(30,32)(31,37)(33,34)(38,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,23)(16,24)(19,22)(25,30,31,26)(27,35,37,32)(28,38,39,33)(29,40,36,34)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10157495);