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 | |
4533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=gi,f^2=gh,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, 10154791);