Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124832 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1555 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,13,16,11,15,19,9,10,14,18,8,12,17,20);(1,2,3,4)(6,10)(7,14)(11,12)(15,17);(6,11)(7,15)(8,16)(10,12)(13,18)(14,17)Polycyclic presentation:
2^9|a^2=f,b^2=dh,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,f^b=fi,e^c=eh,f^c=fghi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41836);