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 | |
4484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,19)(12,15)(13,17)(14,24)(16,20)(18,21)(22,23)(26,31)(29,39)(32,38)(33,40);(2,6)(7,8)(10,12)(11,20)(13,22)(18,24)(27,35)(28,37)(32,38)(33,40);(9,11)(10,16)(12,19)(13,21)(14,22)(15,20)(17,24)(18,23)(25,26,30,31)(27,32,35,38)(28,33,37,40)(29,34,39,36);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,19,20,16)(13,23,22,17)(14,24,21,18)(25,27)(26,32)(28,36)(29,33)(30,35)(31,38)(34,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,13,15,22)(10,17,12,23)(11,21,20,14)(16,24,19,18)(25,28,30,37)(26,33,31,40)(27,36,35,34)(29,38,39,32);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,18,12,24)(11,22,20,13)(16,23,19,17)(25,29)(26,34)(27,33)(28,38)(30,39)(31,36)(32,37)(35,40)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000425);