Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150405 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(2,6,7,8)(9,10,11,15)(12,18,19,24)(13,16,22,23)(14,17,20,21);(1,2)(3,6)(4,8)(5,7)(9,10,13,16,14,17,19,24)(11,18,12,21,20,23,22,15);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,21)(18,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi,e^d=ehi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57719);