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 | #42595 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4415 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,22)(13,18)(14,19)(15,16)(20,21)(23,24)(25,26,32,36)(27,37,39,33)(28,30,29,34)(31,35,38,40);(1,2,4,6)(3,5,8,7)(10,16)(13,19)(17,22)(20,23)(25,27)(26,33)(28,35)(29,40)(30,31)(32,39)(34,38)(36,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,17,16,22)(13,20,19,23)(14,24,18,21)(25,28,32,29)(26,34,36,30)(27,35,39,40)(31,33,38,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,17,16,22)(13,23,19,20)(14,24,18,21)(25,29,32,28)(26,30,36,34)(27,35,39,40)(31,33,38,37);(9,13,15,19)(10,18,16,14)(11,20,12,23)(17,21,22,24)(25,30)(26,28)(27,38)(29,36)(31,39)(32,34)(33,40)(35,37);(9,14,15,18)(10,19,16,13)(11,21,12,24)(17,20,22,23)(25,31,32,38)(26,35,36,40)(27,30,39,34)(28,37,29,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=gi,e^2=ghi,f^2=gi,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, 8177805);