Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144648 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10392 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 793 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,6,8,5,9,4,13)(3,10,14,15,11,16,12,7);(1,3,4,12,5,11,6,14)(2,7,8,10,9,15,13,16);(3,11)(4,6)Polycyclic presentation:
2^9|a^2=dg,b^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72227466 |
| maximal normal subgroup of | #72227466 |
| maximal quotient of | #72238972, #72238975, #72268860 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51844);