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 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24);(2,3)(6,7)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40);(1,3,5,2)(4,7,8,6)(10,14)(12,21)(13,20)(16,19)(26,30)(27,33)(28,35)(39,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,27,29,33)(26,31,30,37)(28,36,35,34)(32,40,38,39);(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28,29,35)(26,32,30,38)(27,34,33,36)(31,39,37,40);(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8916405);