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 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,18,17,14)(12,20,22,16)(19,21,23,24)(26,30)(28,35)(31,36)(34,39);(1,2)(3,8)(4,5)(6,7)(10,13)(11,17)(16,20)(19,23)(27,33)(31,36)(34,39)(38,40);(2,7)(3,4)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,6,4)(2,8,7,5)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24);(1,4,6,3)(2,5,7,8)(9,13,15,10)(11,18,17,14)(12,20,22,16)(19,24,23,21)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,5)(2,3)(4,7)(6,8)(9,14,15,18)(10,17,13,11)(12,21,22,24)(16,23,20,19)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8705700);