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 | |
4631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,14,16)(11,17,12,15)(13,18,21,23)(19,24,20,22)(25,26,31,33)(27,32,30,37)(28,34,38,39)(29,35,40,36);(1,2,5,8)(3,6,4,7)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,24,23,22)(25,27,31,30)(26,32,33,37)(28,36,38,35)(29,39,40,34);(1,3,5,4)(2,6,8,7)(10,16)(15,17)(18,23)(22,24)(26,33)(27,30)(34,39)(35,36);(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,28)(26,34)(27,36)(29,32)(30,35)(31,38)(33,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29)(26,35)(27,34)(28,37)(30,39)(31,40)(32,38)(33,36);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,30,31,27)(26,32,33,37)(28,35,38,36)(29,39,40,34)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=gh,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, 8806103);