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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,21,22,16)(12,13,19,18)(14,24,23,20)(25,26,31,36)(27,32,38,29)(28,34,40,33)(30,39,37,35);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,23)(13,24)(14,19)(15,22)(17,21)(18,20)(25,27,31,38)(26,32,36,29)(28,37,40,30)(33,39,34,35);(1,2)(3,5)(4,6)(7,8)(10,17)(11,22)(12,19)(20,24)(25,28,31,40)(26,33,36,34)(27,30,38,37)(29,39,32,35);(9,12,15,19)(10,18,17,13)(11,23,22,14)(16,20,21,24)(25,28)(26,34)(27,37)(29,39)(30,38)(31,40)(32,35)(33,36);(1,2)(3,7)(4,6)(5,8)(9,13,15,18)(10,19,17,12)(11,20,22,24)(14,16,23,21)(25,29)(26,27)(28,39)(30,33)(31,32)(34,37)(35,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,14,15,23)(10,20,17,24)(11,19,22,12)(13,16,18,21)(25,30,31,37)(26,35,36,39)(27,28,38,40)(29,34,32,33)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9696061);