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 | |
4225 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)(4,5)(6,8)(9,10,15,17)(11,16,21,24)(12,18,20,14)(13,23,22,19)(25,26)(27,33)(28,34)(29,37)(30,31)(32,38)(35,36)(39,40);(2,5)(6,7)(9,11)(10,16)(12,22)(13,20)(14,23)(15,21)(17,24)(18,19)(26,31)(27,34)(32,37)(35,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,21)(12,20)(19,23)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,19,24,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,13,15,22)(10,19,17,23)(11,20,21,12)(14,16,18,24)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40);(9,14,15,18)(10,20,17,12)(11,19,21,23)(13,16,22,24)(25,29,30,38)(26,32,31,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=gi,d^2=gi,e^2=gi,f^2=h,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, 8664782);