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 | |
4468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,14)(13,16)(15,18)(19,24)(20,22)(21,23)(26,30)(31,37)(32,38)(39,40);(2,5)(6,7)(10,12)(11,18)(16,20)(19,21)(26,30)(27,33)(32,38)(34,35);(1,2)(3,6)(4,5)(7,8)(10,12)(15,17)(16,20)(23,24)(25,26,29,30)(27,31,33,37)(28,32,36,38)(34,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(1,3,4,8)(2,6,5,7)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541363);