Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | #392 | 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 | |
425 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 425 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,13,16,14,19,21,18)(11,20,23,37,24,34,39,38)(12,25,29,36,27,17,28,33)(15,30,31,26,32,40,35,22);(1,2,6,3)(4,8,5,7)(9,11,21,39,14,24,13,23)(10,15,18,35,19,32,16,31)(12,26,28,30,27,22,29,40)(17,34,36,37,25,20,33,38);(1,3,6,2)(4,8,5,7)(10,16,19,18)(12,27)(15,31,32,35)(17,33,25,36)(20,37,34,38)(22,26)(28,29)(30,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=gh,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, 43452);