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 | #35212 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,18)(13,22)(14,17)(15,19)(20,23)(21,24)(26,30)(27,33)(32,37)(34,39);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,19,13)(12,20,14,21)(17,24,18,23)(26,30)(31,36)(32,37)(38,40);(9,12,15,14)(10,17,19,18)(11,20,22,21)(13,23,16,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2,5,4)(3,6,8,7)(9,13)(10,11)(12,23)(14,24)(15,16)(17,20)(18,21)(19,22)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4,5,2)(3,7,8,6)(9,10)(11,16)(12,17)(13,22)(14,18)(15,19)(20,24)(21,23);(9,14,15,12)(10,18,19,17)(11,21,22,20)(13,24,16,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553125);