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 | |
4424 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)(4,7)(5,6)(9,10)(11,15)(12,21)(13,16)(14,18)(17,20)(19,24)(22,23)(25,26,30,33)(27,31,37,34)(28,29,32,38)(35,36,39,40);(1,2,6,5)(3,7,4,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,18)(21,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,6,4)(2,7,5,8)(9,12,14,13)(10,16,18,21)(11,19,20,22)(15,23,17,24)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(1,2,6,5)(3,7,4,8)(9,13,14,12)(10,16,18,21)(11,19,20,22)(15,24,17,23)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,4,6,3)(2,7,5,8)(9,10)(11,17)(12,16)(13,21)(14,18)(15,20)(19,24)(22,23)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40);(1,5,6,2)(3,7,4,8)(9,11)(10,17)(12,22)(13,19)(14,20)(15,18)(16,23)(21,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805719);