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 | #2464 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,19)(11,20,23,16)(12,21,22,17)(13,14,18,24)(25,26,32,29)(27,30,33,28)(31,37,39,34)(35,36,40,38);(2,5)(6,7)(9,11,15,23)(10,16,19,20)(12,14,22,24)(13,17,18,21)(25,27)(26,30)(28,29)(31,35)(32,33)(34,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(25,28,32,30)(26,33,29,27)(31,36,39,38)(34,40,37,35);(1,3,4,8)(2,6,5,7)(9,12,15,22)(10,17,19,21)(11,14,23,24)(13,16,18,20)(25,29,32,26)(27,28,33,30)(31,37,39,34)(35,36,40,38);(9,13)(10,14)(11,21)(12,16)(15,18)(17,23)(19,24)(20,22)(25,30,32,28)(26,33,29,27)(31,36,39,38)(34,35,37,40);(9,14)(10,18)(11,22)(12,23)(13,19)(15,24)(16,21)(17,20)(25,31,32,39)(26,34,29,37)(27,35,33,40)(28,36,30,38)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,e^2=h,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, 9701454);