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 | |
4485 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)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26,31,33)(27,32,37,40)(28,34,36,30)(29,39,38,35);(2,3)(6,7)(10,12)(13,21)(15,17)(19,24)(25,27)(26,32)(28,38)(29,36)(30,39)(31,37)(33,40)(34,35);(1,3,5,2)(4,7,8,6)(11,18)(15,17)(19,24)(22,23)(26,33)(27,37)(28,36)(35,39);(1,4,5,8)(2,6,3,7)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28,31,36)(26,34,33,30)(27,38,37,29)(32,35,40,39);(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,29,31,38)(26,35,33,39)(27,36,37,28)(30,32,34,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,34)(26,36,33,28)(27,35,37,39)(29,32,38,40)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664563);