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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,17)(13,14)(15,16)(19,24)(20,23)(21,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,16)(13,19)(17,18)(20,22)(26,31)(27,33)(34,37)(35,40);(9,11,15,12)(10,17,16,18)(13,20,19,22)(14,21,24,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,18,16,17)(13,22,19,20)(14,23,24,21)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,13,15,19)(10,14,16,24)(11,20,12,22)(17,21,18,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,14)(10,19)(11,21)(12,23)(13,16)(15,24)(17,22)(18,20)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549123);