Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | C4×#124 | C22 |
| socle | C22 | #1390 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18,21,25)(19,22,28,31)(20,26,29,24)(23,32,30,27);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19,22,27)(18,23,26,29)(20,25,30,24)(21,28,31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=fi,e^2=fi,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, 1582);