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 | #45445 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,12,13,16)(14,17,23,24)(19,20,21,22)(25,26,32,36)(27,37,39,33)(28,40,38,34)(29,31,30,35);(1,2)(3,7)(4,6)(5,8)(9,11,15,13)(10,16,18,12)(14,19,23,21)(17,22,24,20)(25,27)(26,33)(28,29)(30,38)(31,34)(32,39)(35,40)(36,37);(1,3)(2,5)(4,8)(6,7)(25,28,32,38)(26,34,36,40)(27,29,39,30)(31,37,35,33);(2,6)(5,7)(9,12,15,16)(10,11,18,13)(14,22,23,20)(17,21,24,19)(25,29,32,30)(26,35,36,31)(27,28,39,38)(33,40,37,34);(1,3)(2,7)(4,8)(5,6)(9,13,15,11)(10,16,18,12)(14,19,23,21)(17,20,24,22)(25,30,32,29)(26,31,36,35)(27,38,39,28)(33,34,37,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,22)(18,24)(25,31)(26,30)(27,34)(28,37)(29,36)(32,35)(33,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8596368);