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 | |
4357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(28,35)(32,36)(34,39)(37,40);(3,7)(5,8)(10,12)(13,21)(15,17)(19,24)(25,26)(27,29)(28,36)(30,31)(32,35)(33,38)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(11,18)(15,17)(19,24)(22,23)(26,31)(27,33)(32,36)(34,39);(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27,30,33)(26,29,31,38)(28,34,35,39)(32,37,36,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,28,30,35)(26,32,31,36)(27,34,33,39)(29,37,38,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,33)(27,31)(28,37)(30,38)(32,39)(34,36)(35,40)Polycyclic presentation:
2^9|a^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664691);