Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C4×C8 |
| Frattini, cosocle | C4×#124 | C22 |
| socle | C22 | #1390 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 264 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 — 2 generators on 24 points:
(1,2,4,7,6,8,5,3)(10,14,15,23,17,19,16,24)(11,18,20,21);(1,3,5,8,6,7,4,2)(9,10)(11,19)(12,15)(13,17)(14,20)(16,22)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=gh,f^2=i,b^a=bc,c^a=cf,f^a=fh,c^b=ci,d^b=df,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1340);