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 | #52343 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,17,12,13)(14,18,23,24)(19,22,20,21)(26,32)(28,33)(34,40)(35,38);(1,2)(3,6)(4,7)(5,8)(10,16)(11,12)(18,24)(19,20)(25,26,31,32)(27,33,37,28)(29,34,36,40)(30,35,39,38);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,13,16,17)(14,19,23,20)(18,21,24,22)(25,27,31,37)(26,33,32,28)(29,30,36,39)(34,35,40,38);(9,12,15,11)(10,17,16,13)(14,20,23,19)(18,22,24,21)(25,28)(26,27)(29,38)(30,34)(31,33)(32,37)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,17)(10,11,16,12)(14,22,23,21)(18,20,24,19)(25,29)(26,34)(27,30)(28,38)(31,36)(32,40)(33,35)(37,39);(1,2)(3,5)(4,7)(6,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,30)(26,35)(27,36)(28,34)(29,37)(31,39)(32,38)(33,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003863);