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 | |
4459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,14)(12,15)(13,16)(17,18)(19,24)(20,23)(21,22)(25,26)(27,33)(28,29)(30,34)(31,40)(32,36)(35,37)(38,39);(1,3,6,5)(2,4,8,7)(9,11)(10,14)(12,18)(13,21)(15,17)(16,22)(19,23)(20,24)(25,27)(26,33)(28,35)(29,37)(30,40)(31,34)(32,38)(36,39);(1,4)(2,5)(3,8)(6,7)(10,12)(11,17)(16,20)(19,21)(25,28,32,30)(26,34,36,29)(27,35,38,40)(31,39,37,33);(1,3,6,5)(2,7,8,4)(9,12,15,10)(11,18,17,14)(13,20,23,16)(19,22,21,24)(25,29,32,34)(26,28,36,30)(27,37,38,31)(33,35,39,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,23)(10,16,12,20)(11,19,17,21)(14,24,18,22)(25,30,32,28)(26,34,36,29)(27,35,38,40)(31,33,37,39);(1,5,6,3)(2,7,8,4)(9,14,15,18)(10,17,12,11)(13,22,23,24)(16,19,20,21)(25,31,32,37)(26,35,36,40)(27,34,38,29)(28,39,30,33)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 9620961);