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 | |
4367 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)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26)(27,31)(28,32)(29,37)(30,34)(33,39)(35,36)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27,30,32)(26,31,34,28)(29,35,39,40)(33,38,37,36);(1,3)(2,6)(4,7)(5,8)(10,17)(11,12)(18,24)(19,20)(25,26,30,34)(27,28,32,31)(29,33,39,37)(35,36,40,38);(1,2,5,7)(3,6,8,4)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,13,15,22)(10,18,17,24)(11,19,12,20)(14,21,16,23)(25,28)(26,32)(27,34)(29,38)(30,31)(33,35)(36,39)(37,40);(1,4,5,6)(2,3,7,8)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,29,30,39)(26,33,34,37)(27,35,32,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10168704);