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 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,35)(28,33)(29,36)(30,32)(31,38)(34,40)(37,39);(3,8)(5,6)(12,19)(16,22)(18,23)(20,24)(25,27,30,38)(26,31,32,35)(28,34,39,36)(29,37,40,33);(1,3,4,8)(2,5,7,6)(10,15)(11,17)(16,22)(18,23)(26,32)(28,39)(31,35)(34,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28)(26,33)(27,36)(29,35)(30,39)(31,40)(32,37)(34,38);(1,3)(2,6)(4,8)(5,7)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,26)(27,35)(28,33)(29,36)(30,32)(31,38)(34,40)(37,39);(1,2)(3,5)(4,7)(6,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,29)(26,34)(27,37)(28,35)(30,40)(31,39)(32,36)(33,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9965689);