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 | #2456 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,12)(15,18)(16,20)(22,24)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,37)(39,40);(1,2)(3,5)(4,6)(7,8)(11,17)(15,18)(19,23)(22,24)(26,31)(27,34)(33,36)(35,40);(1,2,4,6)(3,5,7,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(3,7)(5,8)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22);(1,3)(2,5)(4,7)(6,8)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8434667);