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 | |
4538 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)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,32,31)(27,33,39,38)(28,35,29,34)(30,36,37,40);(1,2)(3,6)(4,7)(5,8)(25,27)(26,33)(28,30)(29,37)(31,38)(32,39)(34,40)(35,36);(10,15)(13,22)(16,23)(20,24)(25,28,32,29)(26,34,31,35)(27,37,39,30)(33,36,38,40);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,29,32,28)(26,35,31,34)(27,37,39,30)(33,36,38,40);(1,3,5,4)(2,6,8,7)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,30,32,37)(26,36,31,40)(27,28,39,29)(33,35,38,34);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,31,32,26)(27,38,39,33)(28,34,29,35)(30,40,37,36)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7976701);