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 | #37276 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,24)(12,14,19,22)(13,21,23,18)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,2)(3,7)(4,5)(6,8)(9,11,15,20)(10,16,17,24)(12,18,19,21)(13,22,23,14)(26,31)(27,34)(33,36)(35,40);(2,5)(6,7)(10,17)(11,20)(14,22)(18,21)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(9,12,15,19)(10,14,17,22)(11,21,20,18)(13,16,23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,14)(12,16)(15,23)(17,21)(19,24)(20,22)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,13)(12,17)(15,22)(16,21)(18,24)(20,23)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8105375);