Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1867 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,6)(3,7)(4,8)(9,10,13,16,14,19,21,18)(11,20,24,39,25,36,23,38)(12,26,29,37,30,17,28,35)(15,31,32,27,34,40,33,22);(1,2,5,3)(4,7,8,6)(9,11,21,23,14,25,13,24)(10,15,18,33,19,34,16,32)(12,27,28,31,30,22,29,40)(17,36,37,39,26,20,35,38);(1,3,5,2)(4,6,8,7)(10,16,19,18)(15,32,34,33)(17,37,26,35)(20,38,36,39)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=ehi,d^c=dh,e^c=egi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43440);