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 | |
4191 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)(9,10,15,12)(11,18,17,14)(13,16,22,20)(19,24,23,21)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(27,33)(30,37)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,12)(14,18)(16,20)(21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3,4,8)(2,5,6,7)(9,12,15,10)(11,18,17,14)(13,16,22,20)(19,21,23,24);(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839345);