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 | #57120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4263 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)(9,10,16,19)(11,20,15,17)(12,14,13,18)(21,23,22,24)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,3)(2,6)(4,7)(5,8)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(28,36)(32,38)(35,39)(37,40);(1,2,5,4)(3,6,8,7)(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(1,2,5,4)(3,7,8,6)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,28,31,36)(26,32,30,38)(27,35,33,39)(29,37,34,40);(1,4,5,2)(3,6,8,7)(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,3)(2,7)(4,6)(5,8)(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,d^2=h,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8163798);