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 | #38004 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(26,31)(27,34)(33,40)(35,36);(1,2)(3,8)(4,6)(5,7)(10,15)(12,21)(16,19)(18,22)(25,26,30,31)(27,32,34,29)(28,33,38,40)(35,37,36,39);(1,3,5,6)(2,4,7,8)(10,15)(11,20)(17,24)(18,22);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,21)(23,24)(25,27)(26,32)(28,36)(29,31)(30,34)(33,39)(35,38)(37,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,28,30,38)(26,33,31,40)(27,35,34,36)(29,39,32,37);(1,4)(2,6)(3,7)(5,8)(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,36,40,35)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,e^2=gh,f^2=g,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, 10010642);