Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | C2×#391 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G | |
| LECS | G ▷ C2×#391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14,17,22)(15,18,25,28)(16,21,27,26)(19,24,20,23);(6,9,11,10)(13,15,18,23,17,25,28,24)(14,20,21,27,22,19,26,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,h^a=hi,c^b=cg,g^b=gh,h^b=hi,e^c=ehi,g^c=gi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1504);