Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1982 |
| prime radical | C23 | #418 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 40 points:
(2,5,7,8)(9,10,14,17,13,20,30,35,15,21,31,36,23,19,12,18)(11,22,27,34,28,37,25,38,29,39,24,16,26,33,40,32);(1,2,6,5,4,7,3,8)(9,11,23,28,15,29,13,26)(10,16,19,32,21,34,20,38)(12,24,31,40,30,27,14,25)(17,22,18,37,36,39,35,33);(9,12,23,31,15,30,13,14)(10,17,20,35,21,36,19,18)(11,24,28,40,29,27,26,25)(16,22,38,33,34,39,32,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,f^b=fg,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48046);