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, prime radical, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,23);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,12,16,20)(13,17,22,23)(18,19,24,21)(26,30)(31,37)(32,38)(39,40);(1,2,5,4)(3,6,8,7)(10,15)(11,16)(17,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(9,11,14,16)(10,12,15,20)(13,18,22,24)(17,19,23,21);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,15)(13,21)(14,20)(17,18)(19,22)(23,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,13,14,22)(10,17,15,23)(11,18,16,24)(12,19,20,21)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=gi,f^2=i,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, 7795322);