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 | |
4370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,21)(13,16)(14,22)(15,17)(19,20)(23,24)(26,32)(30,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(25,26,31,32)(27,36,38,33)(28,34,37,30)(29,35,39,40);(9,11,15,13)(10,16,17,18)(12,19,22,23)(14,20,21,24)(25,27,31,38)(26,33,32,36)(28,29,37,39)(30,35,34,40);(9,12)(10,14)(11,19)(13,23)(15,22)(16,20)(17,21)(18,24)(25,28)(26,34)(27,29)(30,32)(31,37)(33,40)(35,36)(38,39);(9,13,15,11)(10,16,17,18)(12,19,22,23)(14,24,21,20)(25,29)(26,35)(27,37)(28,38)(30,36)(31,39)(32,40)(33,34);(1,3,4,7)(2,6,5,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,30,31,34)(26,28,32,37)(27,35,38,40)(29,36,39,33)Polycyclic presentation:
2^9|b^2=i,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553006);