Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #31071 | C2 |
| LCS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(2,3)(5,6)(7,17)(8,14)(9,12)(10,15)(11,16)(13,18)(19,20);(1,2)(3,4)(5,6)(7,15)(8,19)(9,13)(10,17)(11,16)(12,18)(14,20);(1,3,4,2)(5,7,11,10)(6,12,16,19)(8,20,18,9)(13,15,14,17)Polycyclic presentation:
2^9|c^2=di,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,h^a=hi,c^b=cf,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60475);