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 | |
4470 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)(11,19,20,16)(12,17,22,21)(13,23,24,18)(25,26,31,33)(27,32,29,35)(28,34,39,38)(30,37,40,36);(1,2,6,5)(3,8,7,4)(10,15)(12,22)(16,19)(18,23)(25,27)(26,32)(28,37)(29,31)(30,38)(33,35)(34,40)(36,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,20)(12,22)(13,24)(26,33)(27,29)(28,39)(30,40);(1,2,6,5)(3,4,7,8)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,28,31,39)(26,34,33,38)(27,37,29,36)(30,32,40,35);(1,4,6,8)(2,7,5,3)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,29,31,27)(26,35,33,32)(28,37,39,36)(30,34,40,38);(1,5,6,2)(3,8,7,4)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,30,31,40)(26,36,33,37)(27,38,29,34)(28,32,39,35)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,e^c=egi,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, 8914271);