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 | #56310 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,15)(12,18)(13,17)(14,16)(19,22)(20,21)(23,24)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,18)(10,15,16,12)(13,19,23,21)(17,22,24,20)(28,36)(32,39)(35,40)(37,38);(1,3)(2,5)(4,8)(6,7)(10,16)(11,18)(17,24)(19,21)(25,27,30,33)(26,31,34,29)(28,35,36,40)(32,38,39,37);(3,8)(5,6)(9,11)(10,12)(13,21)(14,18)(15,16)(17,22)(19,23)(20,24)(25,28,30,36)(26,32,34,39)(27,35,33,40)(29,37,31,38);(1,2)(3,5)(4,7)(6,8)(9,12)(10,11)(13,22)(14,15)(16,18)(17,21)(19,24)(20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,17,16,24)(11,19,18,21)(12,20,15,22)(25,29,30,31)(26,33,34,27)(28,37,36,38)(32,40,39,35)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653115);