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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,18)(11,16,19,24)(12,22,21,14)(13,23,20,17)(25,26,31,33)(27,32,36,40)(28,29,39,34)(30,38,37,35);(2,3)(6,7)(9,11)(10,16)(12,13)(14,17)(15,19)(18,24)(20,21)(22,23)(25,27)(26,32)(28,38)(29,37)(30,34)(31,36)(33,40)(35,39);(1,3,5,2)(4,7,8,6)(11,19)(12,21)(14,22)(16,24)(26,33)(27,36)(28,39)(30,37);(1,4,5,8)(2,6,3,7)(9,12)(10,14)(11,20)(13,19)(15,21)(16,23)(17,24)(18,22)(25,28,31,39)(26,34,33,29)(27,35,36,38)(30,32,37,40);(9,13,15,20)(10,17,18,23)(11,21,19,12)(14,16,22,24)(25,29,31,34)(26,28,33,39)(27,37,36,30)(32,38,40,35);(9,14)(10,12)(11,17)(13,16)(15,22)(18,21)(19,23)(20,24)(25,30,31,37)(26,35,33,38)(27,34,36,29)(28,32,39,40)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842047);