Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C16 |
| Frattini, cosocle | #1984 | C22 |
| socle | C22 | #1326 |
| prime radical | C23 | #324 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1984 ◁ G | |
| LECS | G ▷ #1984 ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,20,23)(19,26,28,32)(21,24,30,31)(22,25,29,27);(17,18,21,24,22,25,28,32)(19,27,29,31,30,23,20,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cgi,d^b=de,g^b=gi,d^c=dgi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 929);