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 | #45196 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4206 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)(3,5)(4,7)(6,8)(10,12)(15,18)(16,20)(22,24)(26,31)(27,34)(28,36)(37,39);(1,3)(2,4)(5,8)(6,7)(11,17)(15,18)(19,23)(22,24)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,4,6,5)(2,7,8,3)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(2,8)(4,5)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,5,6,4)(2,3,8,7)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066319);