Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #61087 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,23)(12,18)(13,19)(14,21)(15,20)(16,22)(17,25)(24,38)(26,34)(27,33)(28,40)(29,37)(30,36)(31,35)(32,39);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,16)(13,28)(14,25)(15,27)(18,21)(19,34)(20,23)(22,33)(24,29)(26,31)(30,39)(32,35)(36,38)(37,40);(2,7)(3,4)(6,8)(9,12,16,11)(10,18,22,23)(13,26,30,24)(14,25,15,27)(17,20,33,21)(19,34,36,38)(28,29,39,31)(32,35,40,37);(9,13)(10,19)(11,24)(12,26)(14,29)(15,31)(16,30)(17,32)(18,34)(20,35)(21,37)(22,36)(23,38)(25,39)(27,28)(33,40);(9,14,16,15)(10,20,22,21)(11,25,12,27)(13,29,30,31)(17,23,33,18)(19,35,36,37)(24,39,26,28)(32,38,40,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268124);