Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1201 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1201 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,20)(12,17)(13,18)(14,23)(15,19)(16,21)(22,24);(3,8)(9,11,15,12)(10,16,19,24)(13,21,14,22)(17,23,20,18);(2,5)(3,8)(6,7)(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,23,24,18);(9,13,15,14)(10,18,19,23)(11,21,12,22)(16,17,24,20)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394759);