Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #1790 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
636 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,19,14,24)(12,18,13,23)(15,20)(16,21)(17,22);(1,3,6,2)(4,7,5,8)(9,11,15,14)(10,16,20,17)(12,21,13,22)(18,24,23,19);(9,12)(10,17,20,16)(11,21)(13,15)(14,22)(18,24,23,19)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38680);