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 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,31,30)(27,35,37,32)(28,38,39,33)(29,40,36,34);(1,2)(3,7)(4,8)(5,6)(25,27,31,37)(26,32,30,35)(28,36,39,29)(33,34,38,40);(10,11)(15,17)(16,18)(22,23)(25,28,31,39)(26,33,30,38)(27,29,37,36)(32,40,35,34);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,33)(27,36)(29,37)(30,38)(31,39)(32,34)(35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29,31,36)(26,34,30,40)(27,28,37,39)(32,38,35,33);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,26)(27,35,37,32)(28,38,39,33)(29,34,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=hi,e^2=g,f^2=ghi,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, 7847138);