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 | |
4420 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)(11,21,23,16)(12,13,19,18)(14,22,24,20)(25,26)(27,32)(28,33)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,23)(10,16,17,21)(12,14,19,24)(13,20,18,22)(25,27,31,33)(26,32,35,28)(29,36,34,40)(30,37,39,38);(2,6)(3,7)(10,17)(13,18)(16,21)(20,22)(25,28,31,32)(26,27,35,33)(29,38,34,37)(30,36,39,40);(3,7)(4,8)(9,12)(10,18)(11,14)(13,17)(15,19)(16,22)(20,21)(23,24)(25,29)(26,30)(27,36)(28,38)(31,34)(32,37)(33,40)(35,39);(1,3)(2,4)(5,7)(6,8)(9,13,15,18)(10,19,17,12)(11,22,23,20)(14,21,24,16)(25,28)(26,33)(27,35)(29,37)(30,36)(31,32)(34,38)(39,40);(1,4)(2,3)(5,8)(6,7)(9,14)(10,20)(11,19)(12,23)(13,21)(15,24)(16,18)(17,22)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8971179);