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 |
| prime radical | #52459 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,16)(12,21)(13,17)(14,20)(15,22)(18,23)(19,24)(25,26,31,29)(27,32,35,28)(30,37,39,33)(34,36,38,40);(1,3)(2,7)(4,5)(6,8)(9,11)(10,14)(12,18)(13,20)(15,23)(16,17)(19,22)(21,24)(25,27,31,35)(26,32,29,28)(30,34,39,38)(33,40,37,36);(1,2,6,4)(3,7,8,5)(9,12)(10,15)(11,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,28,31,32)(26,27,29,35)(30,36,39,40)(33,34,37,38);(2,4)(3,8)(9,12)(10,15)(11,19)(13,22)(14,24)(16,23)(17,21)(18,20)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,40,38,36);(1,4,6,2)(3,7,8,5)(9,11)(10,16)(12,18)(13,20)(14,17)(15,24)(19,22)(21,23)(25,27)(26,32)(28,29)(30,38)(31,35)(33,36)(34,39)(37,40);(1,5,6,7)(2,3,4,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,23)(19,24)(21,22)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227614);