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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,33)(27,32,37,39)(28,34,38,40)(29,35,36,30);(1,2,6,5)(3,8,7,4)(10,15)(12,17)(16,21)(20,23)(25,27)(26,32)(28,36)(29,38)(30,34)(31,37)(33,39)(35,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(13,22)(20,23)(26,33)(27,37)(28,38)(30,35);(1,2,6,5)(3,4,7,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28,31,38)(26,34,33,40)(27,36,37,29)(30,39,35,32);(1,4,6,8)(2,7,5,3)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,36)(26,35,33,30)(27,28,37,38)(32,34,39,40);(1,5,6,2)(3,8,7,4)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30,31,35)(26,36,33,29)(27,34,37,40)(28,39,38,32)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8827922);