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 | |
4394 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)(3,8)(4,5)(6,7)(9,10,15,19)(11,16,20,23)(12,17,21,22)(13,14,18,24)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,39)(37,38);(1,3,6,5)(2,4,7,8)(9,11,15,20)(10,16,19,23)(12,13,21,18)(14,22,24,17)(25,27,31,35)(26,32,33,30)(28,38,34,36)(29,37,40,39);(1,3,6,5)(2,4,7,8)(11,20)(12,21)(16,23)(17,22)(26,33)(27,35)(28,34)(37,39);(2,7)(3,5)(9,12,15,21)(10,17,19,22)(11,13,20,18)(14,23,24,16)(25,28,31,34)(26,29,33,40)(27,36,35,38)(30,37,32,39);(1,4,6,8)(2,3,7,5)(9,13)(10,14)(11,21)(12,20)(15,18)(16,22)(17,23)(19,24)(25,29)(26,34)(27,37)(28,33)(30,38)(31,40)(32,36)(35,39);(1,5,6,3)(2,4,7,8)(9,14)(10,18)(11,22)(12,16)(13,19)(15,24)(17,20)(21,23)(25,30,31,32)(26,35,33,27)(28,39,34,37)(29,36,40,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=g,d^2=i,e^2=g,f^2=h,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, 8625172);