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 | #53559 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,17,22,21)(19,20,24,23)(25,26,32,36)(27,37,39,33)(28,35,30,34)(29,38,40,31);(1,2)(3,8)(4,6)(5,7)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,27,32,39)(26,33,36,37)(28,38,30,31)(29,34,40,35);(1,3,5,6)(2,4,7,8)(25,28,32,30)(26,34,36,35)(27,38,39,31)(29,33,40,37);(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,29,32,40)(26,31,36,38)(27,35,39,34)(28,33,30,37);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,30,32,28)(26,35,36,34)(27,31,39,38)(29,37,40,33);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,31)(26,29)(27,28)(30,39)(32,38)(33,34)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332083);