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 | |
4552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26)(27,31)(28,32)(29,37)(30,34)(33,39)(35,36)(38,40);(2,3)(6,7)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,27,30,32)(26,31,34,28)(29,35,39,40)(33,38,37,36);(1,3,5,2)(4,7,8,6)(10,17)(11,20)(12,22)(18,24)(25,26,30,34)(27,28,32,31)(29,33,39,37)(35,36,40,38);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,28,30,31)(26,32,34,27)(29,38,39,36)(33,35,37,40);(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,e^2=gi,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, 9746464);