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 | #53893 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,22)(10,16,20,21)(12,13,19,23)(14,18,17,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2)(3,8)(4,7)(5,6)(25,27,29,35)(26,31,33,30)(28,34,37,40)(32,39,36,38);(1,3,5,4)(2,7,6,8)(9,12,15,19)(10,17,20,14)(11,13,22,23)(16,24,21,18)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,4,5,3)(2,7,6,8)(9,13)(10,18)(11,19)(12,22)(14,21)(15,23)(16,17)(20,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,14,15,17)(10,19,20,12)(11,18,22,24)(13,16,23,21)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9636407);