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 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,18)(13,16)(14,15)(19,22)(20,21)(23,24)(25,26,30,31)(27,29,33,39)(28,35,37,32)(34,36,40,38);(1,2)(3,6)(4,7)(5,8)(10,15)(11,18)(16,24)(19,21)(25,27,30,33)(26,29,31,39)(28,34,37,40)(32,38,35,36);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(16,24)(19,21)(26,31)(27,33)(32,35)(34,40);(3,8)(5,6)(9,11)(10,12)(13,21)(14,18)(15,17)(16,22)(19,23)(20,24)(25,27)(26,29)(28,34)(30,33)(31,39)(32,38)(35,36)(37,40);(1,2)(3,5)(4,7)(6,8)(9,12)(10,11)(13,22)(14,17)(15,18)(16,21)(19,24)(20,23)(25,28,30,37)(26,32,31,35)(27,34,33,40)(29,38,39,36);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,16,15,24)(11,19,18,21)(12,20,17,22)(25,29)(26,33)(27,31)(28,36)(30,39)(32,34)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8610143);