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 | |
4299 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)(9,10,15,16)(11,14,17,24)(12,13,21,22)(18,23,20,19);(2,3)(6,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,26)(27,30)(28,31)(29,32)(33,38)(34,39)(35,37)(36,40);(1,3,5,2)(4,7,8,6)(10,16)(11,17)(12,21)(19,23)(25,26,29,32)(27,30,34,39)(28,31,37,35)(33,38,36,40);(1,4,5,8)(2,6,3,7)(9,12)(10,13)(11,18)(14,23)(15,21)(16,22)(17,20)(19,24)(25,26)(27,30)(28,35)(29,32)(31,37)(33,40)(34,39)(36,38);(9,13)(10,12)(11,19)(14,20)(15,22)(16,21)(17,23)(18,24)(25,27,29,34)(26,30,32,39)(28,36,37,33)(31,40,35,38);(9,14,15,24)(10,17,16,11)(12,19,21,23)(13,18,22,20)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633682);