Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12525 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,16,14,20,15,18)(11,21,24,37,22,38,26,36)(13,27,23,35,28,33,29,19)(17,31,30,39,34,40,32,25);(1,3,7,2,5,8,4,6)(10,16,20,18)(11,22)(17,32,34,30)(19,27,35,33)(21,36,38,37)(24,26)(25,39)(31,40);(2,6)(3,8)(9,11,23,31,15,26,13,25,14,22,29,40,12,24,28,39)(10,17,33,36,16,30,19,21,20,34,27,37,18,32,35,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48924);