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 | |
4335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,16)(11,14,19,24)(12,17,21,23)(13,22,20,18)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,18,23,22)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(10,16)(11,19)(12,21)(18,22)(26,33)(28,34)(32,36)(35,40);(9,12)(10,17)(11,20)(13,19)(14,18)(15,21)(16,23)(22,24)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(1,3,5,4)(2,6,8,7)(9,13,15,20)(10,18,16,22)(11,12,19,21)(14,23,24,17)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(1,4,5,3)(2,7,8,6)(9,14)(10,19)(11,16)(12,18)(13,17)(15,24)(20,23)(21,22)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=hi,e^2=gh,f^2=g,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, 7776114);