Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150407 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C23 | C2×#99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 56 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)(9,10,13,18,15,11,20,22)(12,17,14,19,24,21,16,23);(1,2)(3,6)(4,8)(5,7)(9,11,14,23,15,10,16,19)(12,22,20,21,24,18,13,17);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,21)(13,14)(15,24)(16,20)(18,19)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfi,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, 57724);