Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94530 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7339 |
| commutator | C2×C22⋊C4 | C2×C8 |
| Frattini, cosocle | #1819 | C22 |
| prime radical | #3490 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1819 ◁ G | |
| LECS | G ▷ #1819 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,8)(4,7,12,14,13,16,10,6)(5,9,11,15);(1,3)(2,6)(4,10)(5,11)(7,15)(8,16)(9,14)(12,13)Polycyclic presentation:
2^9|a^2=d,c^2=h,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=deh,f^b=fh,g^b=gi,d^c=dg,e^c=ei,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1726);