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 | |
4462 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)(12,20)(16,23)(18,24)(21,22)(25,26,30,33)(27,34,37,31)(28,38,29,32)(35,39,36,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,34,40);(1,4,5,3)(2,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,38)(30,33)(34,37)(35,39)(36,40);(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,29,30,28)(26,32,33,38)(27,36,37,35)(31,39,34,40);(9,12,13,20)(10,16,17,23)(11,18,19,24)(14,21,15,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7554684);