Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139532 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #23984 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,4,8,5,9,3,7)(6,10,12,15,13,16,11,14);(2,7,9,8)(10,14,16,15);(3,11)(4,12)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213827, #72216452, #72231770, … |
| maximal normal subgroup of | #72213827, #72216452, #72231770, … |
| maximal quotient of | #72213786, #72213791, #72237739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46818);