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 | |
4412 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)(11,13)(12,19)(14,22)(15,17)(16,23)(18,20)(21,24)(25,26,30,33)(27,31,37,34)(28,29,32,38)(35,36,39,40);(1,2,6,5)(3,7,4,8)(9,11,15,19)(10,13,17,12)(14,18,23,24)(16,21,22,20)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,6,4)(2,7,5,8)(9,12,15,13)(10,11,17,19)(14,21,23,20)(16,24,22,18)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(1,2,6,5)(3,7,4,8)(9,13,15,12)(10,11,17,19)(14,21,23,20)(16,18,22,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,4,6,3)(2,7,5,8)(9,11)(10,13)(12,17)(14,18)(15,19)(16,21)(20,22)(23,24)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40);(1,5,6,2)(3,7,4,8)(9,14)(10,16)(11,18)(12,20)(13,21)(15,23)(17,22)(19,24)(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=h,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, 8805772);