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 | |
4553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,8)(6,7)(9,10)(11,19)(12,21)(13,23)(14,15)(16,20)(17,22)(18,24)(25,26,31,33)(27,32,35,40)(28,30,38,39)(29,37,36,34);(2,7)(3,8)(10,15)(13,24)(16,19)(17,21)(25,27,31,35)(26,32,33,40)(28,36,38,29)(30,34,39,37);(1,3,6,8)(2,4,7,5)(10,15)(11,20)(17,21)(18,23)(26,33)(27,35)(28,38)(34,37);(1,4,6,5)(2,3,7,8)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,24)(22,23)(25,28)(26,30)(27,36)(29,35)(31,38)(32,34)(33,39)(37,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,29,31,36)(26,34,33,37)(27,28,35,38)(30,32,39,40);(1,5,6,4)(2,8,7,3)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,30)(26,28)(27,37)(29,32)(31,39)(33,38)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662652);