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 | |
4332 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)(3,4)(5,8)(6,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,22)(20,24)(25,26)(27,36)(28,33)(29,32)(30,34)(31,35)(37,38)(39,40);(1,2)(3,6)(4,7)(5,8)(10,15)(12,20)(16,22)(18,21)(25,27,31,29)(26,32,35,36)(28,37,34,39)(30,38,33,40);(1,2,5,8)(3,4,7,6)(10,15)(11,19)(17,24)(18,21);(1,3)(2,6)(4,8)(5,7)(9,11,14,19)(10,16,15,22)(12,18,20,21)(13,17,23,24)(25,28,31,34)(26,33,35,30)(27,37,29,39)(32,40,36,38);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,22)(14,20)(15,24)(16,23)(19,21)(25,29,31,27)(26,32,35,36)(28,39,34,37)(30,38,33,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,18,15,21)(11,17,19,24)(12,16,20,22)(25,30,31,33)(26,34,35,28)(27,38,29,40)(32,39,36,37)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9636413);