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 | |
4511 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,29)(27,34,33,30)(28,36,38,32)(35,37,39,40);(1,2)(3,8)(4,6)(5,7)(11,17)(15,21)(18,23)(22,24)(25,27,31,33)(26,30,29,34)(28,35,38,39)(32,37,36,40);(1,3,5,6)(2,4,7,8)(10,14)(15,21)(16,19)(22,24)(25,28,31,38)(26,32,29,36)(27,35,33,39)(30,37,34,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,26)(27,30,33,34)(28,36,38,32)(35,37,39,40);(1,4)(2,6)(3,7)(5,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30)(26,33)(27,29)(28,37)(31,34)(32,39)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,e^2=g,f^2=gh,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, 9590246);