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 | |
4509 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)(11,13)(12,16)(14,15)(17,22)(18,21)(19,24)(20,23)(25,26)(27,31)(28,34)(29,37)(30,32)(33,36)(35,40)(38,39);(1,3,5,8)(2,6,4,7)(10,15)(11,17)(16,23)(18,24)(25,27,30,36)(26,31,32,33)(28,37,39,35)(29,38,40,34);(10,15)(11,17)(16,23)(18,24)(26,32)(27,36)(34,38)(35,37);(1,2,5,4)(3,7,8,6)(9,11,14,17)(10,13,15,22)(12,18,20,24)(16,21,23,19)(25,27)(26,33)(28,37)(29,38)(30,36)(31,32)(34,40)(35,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,28)(26,34)(27,37)(29,31)(30,39)(32,38)(33,40)(35,36);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,29,30,40)(26,35,32,37)(27,38,36,34)(28,33,39,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9844986);