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 | #52343 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(25,26,31,32)(27,34,29,33)(28,37,38,30)(35,40,39,36);(1,2)(3,4)(5,7)(6,8)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(26,32)(28,38)(33,34)(35,39);(1,3,5,8)(2,6,7,4)(10,15)(11,17)(16,23)(18,24)(25,27,31,29)(26,33,32,34)(28,39,38,35)(30,36,37,40);(2,7)(3,8)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28,31,38)(26,30,32,37)(27,35,29,39)(33,40,34,36);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29,31,27)(26,34,32,33)(28,39,38,35)(30,36,37,40);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,38)(33,39)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8616429);