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 | #44698 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4182 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)(3,8)(4,7)(5,6)(9,10,13,15)(11,16,19,14)(12,20,22,17)(18,23,21,24)(26,31)(27,34)(33,40)(35,36);(3,4)(7,8)(9,11,13,19)(10,14,15,16)(12,18,22,21)(17,23,20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,39)(38,40);(1,3,6,4)(2,7,5,8)(10,15)(14,16)(17,20)(23,24);(3,4)(7,8)(9,10)(11,14)(12,17)(13,15)(16,19)(18,23)(20,22)(21,24)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,39,40,37);(1,4,6,3)(2,7,5,8)(9,11)(10,16)(12,21)(13,19)(14,15)(17,23)(18,22)(20,24)(25,28,30,38)(26,33,31,40)(27,35,34,36)(29,39,32,37);(1,5,6,2)(3,8,4,7)(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,35,40,36)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050782);