Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,3,6)(4,7,10,13)(5,8,11,9)(12,14,16,15);(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14);(3,10)(6,13)(9,15)(11,16)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72210587, #72213719, #72215653, … |
| maximal normal subgroup of | #72210587, #72213719, #72215653, … |
| maximal quotient of | #72210539, #72213260, #72213646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46914);