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 | #48109 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4551 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)(11,18)(12,20)(13,14)(15,16)(17,23)(19,22)(21,24)(26,31)(27,33)(32,35)(34,39);(2,5)(6,8)(9,11,15,14)(10,13,16,18)(12,21,23,19)(17,24,20,22)(25,26,30,31)(27,29,33,38)(28,35,36,32)(34,40,39,37);(1,2,4,5)(3,6,7,8)(10,16)(13,18)(17,20)(22,24)(28,36)(32,35)(34,39)(37,40);(3,7)(6,8)(9,12,15,23)(10,17,16,20)(11,19,14,21)(13,22,18,24)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,22)(14,16)(15,18)(17,19)(20,21)(23,24)(25,28,30,36)(26,32,31,35)(27,34,33,39)(29,37,38,40);(1,3)(2,6)(4,7)(5,8)(9,14,15,11)(10,13,16,18)(12,19,23,21)(17,24,20,22)(25,29)(26,33)(27,31)(28,37)(30,38)(32,39)(34,35)(36,40)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8508845);