Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501244 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33582 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(11,21)(12,23)(13,29)(14,28)(16,31)(17,18)(19,30)(22,35)(24,25)(26,38)(27,34)(32,40)(33,39)(36,37);(1,3,5,4)(2,6,8,7)(9,10)(11,22)(12,27)(13,17)(14,30)(15,20)(16,24)(18,29)(19,28)(21,35)(23,34)(25,31)(26,40)(32,38)(33,37)(36,39);(1,3,5,4)(2,7,8,6)(9,11,23,37,14,25,29,38,15,26,13,24,28,36,12,21)(10,16,18,22,19,32,34,39,20,33,27,40,30,35,17,31);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410756);