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 | #38141 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4421 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)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,31,28)(27,35,36,32)(29,33,39,37)(30,38,40,34);(1,3,5,8)(2,6,4,7)(25,27)(26,32)(28,35)(29,30)(31,36)(33,34)(37,38)(39,40);(10,14)(15,17)(16,20)(22,23)(27,36)(29,39)(32,35)(33,37);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,31,26)(27,35,36,32)(29,37,39,33)(30,38,40,34);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,33)(27,30)(28,37)(31,39)(32,34)(35,38)(36,40);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30)(26,34)(27,29)(28,38)(31,40)(32,33)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549806);