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 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,12,17,16)(13,21,22,18)(19,23,24,20);(2,6)(3,7)(10,15)(12,16)(18,21)(20,23)(25,26,31,34)(27,35,29,32)(28,30,33,40)(36,39,38,37);(1,3)(2,4)(5,7)(6,8)(10,15)(11,17)(13,22)(20,23)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,39,40,37);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,37,35,39);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,39,40,37);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,38)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gi,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, 8831902);