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 | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,19,21,16)(12,22,20,17)(14,18,24,23)(25,26)(27,29)(28,33)(30,39)(31,32)(34,36)(35,38)(37,40);(1,2,4,6)(3,5,8,7)(9,11,15,21)(10,16,13,19)(12,14,20,24)(17,18,22,23)(25,27)(26,29)(28,36)(30,37)(31,35)(32,38)(33,34)(39,40);(1,2)(3,5)(4,6)(7,8)(26,32)(27,35)(28,37)(34,39);(9,12,15,20)(10,17,13,22)(11,14,21,24)(16,18,19,23)(25,28,31,37)(26,33,32,40)(27,36,35,30)(29,34,38,39);(1,2)(3,7)(4,6)(5,8)(9,13,15,10)(11,16,21,19)(12,22,20,17)(14,18,24,23)(25,29)(26,27)(28,34)(30,40)(31,38)(32,35)(33,36)(37,39);(1,3)(2,5)(4,8)(6,7)(9,14)(10,18)(11,20)(12,21)(13,23)(15,24)(16,22)(17,19)(25,30,31,36)(26,34,32,39)(27,37,35,28)(29,33,38,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9696363);