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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26,31,33)(27,32,36,30)(28,34,37,40)(29,35,39,38);(1,2)(3,8)(4,6)(5,7)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(25,27,31,36)(26,32,33,30)(28,29,37,39)(34,35,40,38);(1,3,5,6)(2,4,7,8)(26,33)(27,36)(28,37)(35,38);(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24)(25,28)(26,34)(27,29)(30,38)(31,37)(32,35)(33,40)(36,39);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,16,17,18)(12,22,23,19)(14,20,21,24)(25,29)(26,35)(27,28)(30,40)(31,39)(32,34)(33,38)(36,37);(1,4)(2,6)(3,7)(5,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,30,31,32)(26,36,33,27)(28,38,37,35)(29,34,39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,e^2=g,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, 9590571);