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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,13)(14,16)(15,17)(18,22)(20,21)(23,24)(25,26,32,37)(27,35,29,33)(28,34,39,38)(30,31,36,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,19)(12,20,22,23)(13,24,18,21)(25,27,32,29)(26,33,37,35)(28,36,39,30)(31,38,40,34);(1,3,5,6)(2,4,7,8)(10,17)(12,22)(16,19)(20,23)(25,28,32,39)(26,34,37,38)(27,36,29,30)(31,35,40,33);(9,12)(10,18)(11,20)(13,17)(14,23)(15,22)(16,21)(19,24)(25,29,32,27)(26,35,37,33)(28,30,39,36)(31,34,40,38);(1,2)(3,4)(5,7)(6,8)(9,13,15,18)(10,12,17,22)(11,21,14,24)(16,23,19,20)(25,30)(26,31)(27,28)(29,39)(32,36)(33,38)(34,35)(37,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,19,17,16)(12,23,22,20)(13,21,18,24)(25,31)(26,36)(27,38)(28,35)(29,34)(30,37)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9591697);