Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94662 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1404 |
| commutator | #382 | C2×C4 |
| Frattini, cosocle | #1836 | C22 |
| prime radical | C23 | AΣL(1,𝔽16) |
| LCS | G ▷ #382 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1836 ◁ G | |
| LECS | G ▷ #1836 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
138 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 26 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:
(2,5,6,7)(9,10,12,17,14,20,21,24,15,18,22,23,13,19,11,16);(1,2)(3,7)(4,6)(5,8)(10,16,19,17,18,24,20,23)(11,12,21,22)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=fg,d^2=g,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fi,h^a=hi,c^b=cf,d^b=defghi,e^b=eh,f^b=fh,g^b=gh,d^c=dgh,e^c=eh,f^c=fi,g^c=gi,e^d=ei35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1924);