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 | |
4207 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)(11,17)(12,13)(14,15)(16,20)(18,23)(19,24)(21,22)(25,26)(27,29)(28,33)(30,39)(31,32)(34,36)(35,38)(37,40);(1,2,4,6)(3,5,8,7)(10,15)(12,21)(16,17)(18,24)(25,27)(26,29)(28,36)(30,37)(31,35)(32,38)(33,34)(39,40);(1,2)(3,5)(4,6)(7,8)(26,32)(27,35)(28,37)(34,39);(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,28,31,37)(26,33,32,40)(27,36,35,30)(29,34,38,39);(1,2)(3,7)(4,6)(5,8)(9,12,14,21)(10,13,15,22)(11,18,20,24)(16,19,17,23)(25,29)(26,27)(28,34)(30,40)(31,38)(32,35)(33,36)(37,39);(1,3)(2,5)(4,8)(6,7)(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,30,31,36)(26,34,32,39)(27,37,35,28)(29,33,38,40)Polycyclic presentation:
2^9|b^2=hi,d^2=gh,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, 9696360);