Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149889 |
| 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 | C22×Q8 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | #3497 | C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 434 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:
(1,2,4,8,5,6,3,7)(9,10,12,16)(11,14,19,21)(13,18,23,24)(15,20,17,22);(2,6)(3,4)(9,10)(11,14)(12,21)(13,18)(15,23)(16,22)(17,20)(19,24);(1,3,5,4)(2,7,6,8)(9,11,13,20)(10,14,18,17)(12,19,23,22)(15,16,21,24)Polycyclic presentation:
2^9|a^2=e,c^2=gi,d^2=i,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=ef,f^b=fi,h^b=hi,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, 57149);