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 | #2555 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,31,35)(27,36,38,32)(28,29,39,33)(30,34,37,40);(1,2)(3,5)(4,6)(7,8)(25,27,31,38)(26,32,35,36)(28,37,39,30)(29,34,33,40);(10,15)(11,16)(17,22)(18,23)(25,28,31,39)(26,33,35,29)(27,30,38,37)(32,34,36,40);(1,3,4,7)(2,5,6,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28)(26,29)(27,37)(30,38)(31,39)(32,34)(33,35)(36,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29)(26,28)(27,34)(30,36)(31,33)(32,37)(35,39)(38,40);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30,31,37)(26,34,35,40)(27,28,38,39)(29,32,33,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7772262);