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 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(25,26,30,34)(27,31,32,28)(29,33,38,37)(35,39,40,36);(2,4)(6,7)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36);(1,4,5,2)(3,7,8,6)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,36)(39,40);(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,3)(2,7)(4,6)(5,8)(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8374410);