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 | #40044 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(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);(1,2)(3,5)(4,6)(7,8)(25,27)(26,32)(28,35)(29,34)(30,37)(31,36)(33,40)(38,39);(10,14)(15,17)(16,20)(22,23)(27,36)(29,39)(32,35)(33,37);(1,3,4,7)(2,5,6,8)(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)(25,29,31,39)(26,33,28,37)(27,34,36,38)(30,32,40,35);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30)(26,34)(27,37)(28,38)(29,32)(31,40)(33,36)(35,39)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=i,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, 7550176);