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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,14,20,16)(12,22,23,18)(13,24,21,19)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,17,14)(12,21,23,13)(18,19,22,24)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(10,17)(11,20)(12,23)(19,24)(26,33)(28,40)(32,36)(35,37);(9,12)(10,18)(11,21)(13,20)(14,24)(15,23)(16,19)(17,22)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(9,13,15,21)(10,19,17,24)(11,12,20,23)(14,18,16,22)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(1,4,5,2)(3,7,8,6)(9,14,15,16)(10,20,17,11)(12,24,23,19)(13,22,21,18)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7861879);