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 | |
4155 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,20)(22,24)(25,26)(27,36)(28,33)(29,32)(30,34)(31,35)(37,38)(39,40);(1,2,4,6)(3,5,8,7)(10,15)(12,16)(18,21)(20,22)(25,27,31,29)(26,32,35,36)(28,37,34,39)(30,38,33,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,17)(18,21)(19,24)(25,28,31,34)(26,33,35,30)(27,37,29,39)(32,40,36,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,16,15,12)(13,19,23,24)(18,22,21,20);(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,29,31,27)(26,32,35,36)(28,39,34,37)(30,38,33,40);(9,13,14,23)(10,18,15,21)(11,19,17,24)(12,20,16,22)(25,30,31,33)(26,34,35,28)(27,38,29,40)(32,39,36,37)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=h,e^2=gi,f^2=gi,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, 9633457);