Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C2×#124 | C23 |
| socle | C22 | C2×AΣL(1,𝔽16) |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
729 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 729 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,12)(7,10,15,17)(9,14,19,20)(11,16,13,18);(1,2,3,4)(5,6)(7,10)(8,17)(9,14)(11,19)(12,18)(13,16)(15,20);(5,7,9,16)(6,10,14,13)(8,15,19,18)(11,12,17,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bd,d^a=dg,g^a=gh,h^a=hi,d^b=di,e^b=eg,g^b=gi,h^b=hi,e^d=ehi,g^d=gi,g^e=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58998);