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 | |
4203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,13)(12,18,22,24)(14,23,21,19)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,37)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,20)(10,16,17,13)(12,21,22,14)(18,19,24,23)(25,26,31,36)(27,32,39,40)(28,33,37,38)(29,30,35,34);(1,2)(3,5)(4,6)(7,8)(10,17)(11,20)(12,22)(19,23)(25,27,31,39)(26,32,36,40)(28,29,37,35)(30,38,34,33);(9,12)(10,18)(11,21)(13,23)(14,20)(15,22)(16,19)(17,24)(25,28,31,37)(26,33,36,38)(27,29,39,35)(30,40,34,32);(1,2)(3,7)(4,6)(5,8)(9,13)(10,11)(12,23)(14,24)(15,16)(17,20)(18,21)(19,22)(25,29)(26,34)(27,37)(28,39)(30,36)(31,35)(32,33)(38,40);(1,3)(2,5)(4,8)(6,7)(9,14,15,21)(10,19,17,23)(11,12,20,22)(13,18,16,24)(25,30,31,34)(26,35,36,29)(27,38,39,33)(28,40,37,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=g,f^2=gh,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, 9695833);