Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| 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 | |
410 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(2,5)(3,6)(7,8)(9,10,14,20,16,21,15,33)(11,18,23,37,25,38,24,39)(12,26,22,17,29,35,28,36)(13,30,31,34,32,40,27,19);(1,2,6,8,4,7,3,5)(9,11,13,22,14,23,31,29,16,25,32,28,15,24,27,12)(10,17,34,38,33,26,30,37,21,36,19,18,20,35,40,39);(9,12,27,24,15,28,32,25,16,29,31,23,14,22,13,11)(10,18,30,17,20,37,34,35,21,38,40,36,33,39,19,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eh,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48879);