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 | |
4676 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297742);