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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(25,26,30,32)(27,31,35,38)(28,33,37,40)(29,36,39,34);(1,2)(3,5)(4,7)(6,8)(10,11)(15,17)(16,18)(22,23)(25,27,30,35)(26,31,32,38)(28,36,37,34)(29,33,39,40);(1,3)(2,7)(4,6)(5,8)(10,11)(12,19)(13,21)(22,23)(26,32)(27,35)(28,37)(29,39);(2,8)(3,4)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,37)(39,40);(1,4,6,3)(2,5,8,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,40)(34,35);(1,5)(2,3)(4,8)(6,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,39)(26,34,32,36)(27,33,35,40)(28,38,37,31)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970444);