Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1361 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,20,15,22,14,21,17,23,13,32,16,33,31,38)(11,24,26,36,28,37,25,39,30,40,27,19,29,18,34,35);(1,3)(2,5)(4,8)(6,7)(10,18)(11,25)(12,31)(13,14)(15,16)(19,20)(21,39)(22,40)(23,37)(24,33)(26,28)(27,29)(30,34)(32,36)(35,38)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,d^2=g,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1115);