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 |
| prime radical | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,16,22,24)(19,23,20,21)(25,26)(27,37)(28,31)(29,38)(30,34)(32,36)(33,40)(35,39);(2,5)(6,7)(9,11,14,17)(10,15,18,13)(12,20,22,19)(16,23,24,21)(25,27)(26,33)(28,38)(29,34)(30,35)(31,39)(32,40)(36,37);(1,2)(3,6)(4,5)(7,8)(25,28,32,30)(26,34,36,31)(27,38,40,35)(29,37,39,33);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,23)(22,24)(25,29,32,39)(26,35,36,38)(27,34,40,31)(28,33,30,37);(1,2)(3,7)(4,5)(6,8)(9,12,14,22)(10,16,18,24)(11,19,17,20)(13,23,15,21)(25,30,32,28)(26,34,36,31)(27,35,40,38)(29,37,39,33);(1,3)(2,6)(4,8)(5,7)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,19,24,20)(25,31,32,34)(26,30,36,28)(27,39,40,29)(33,35,37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=g,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, 9686684);