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 | |
4669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(11,12)(13,21)(22,23)(26,30)(31,36)(32,37)(38,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,14)(13,18)(15,16)(19,22)(20,23)(21,24)(27,33)(31,36)(34,39)(38,40);(1,3,5,6)(2,4,7,8)(9,11,15,12)(10,17,16,14)(13,19,21,20)(18,22,24,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,4)(5,7)(6,8)(9,12,15,11)(10,17,16,14)(13,20,21,19)(18,22,24,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,21)(10,18,16,24)(11,19,12,20)(14,23,17,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,17)(10,12,16,11)(13,22,21,23)(18,19,24,20)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572581);