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 | #2856 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4344 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)(11,12)(13,21)(14,18)(15,16)(17,23)(19,24)(20,22)(25,26,32,31)(27,37,39,33)(28,40,38,34)(29,36,30,35);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,12,16,18)(13,19,23,22)(17,20,21,24)(25,27)(26,33)(28,29)(30,38)(31,37)(32,39)(34,36)(35,40);(2,6)(4,7)(10,16)(11,14)(17,21)(19,22)(25,28,32,38)(26,34,31,40)(27,29,39,30)(33,36,37,35);(1,3)(2,7)(4,6)(5,8)(9,12,15,18)(10,11,16,14)(13,20,23,24)(17,19,21,22)(25,29,32,30)(26,35,31,36)(27,28,39,38)(33,40,37,34);(9,13,15,23)(10,17,16,21)(11,19,14,22)(12,20,18,24)(25,30,32,29)(26,36,31,35)(27,38,39,28)(33,34,37,40);(1,4,5,7)(2,3,6,8)(9,14,15,11)(10,18,16,12)(13,22,23,19)(17,24,21,20)(25,31,32,26)(27,37,39,33)(28,34,38,40)(29,36,30,35)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8102363);