Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139730 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #23900 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1189 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,6,8,4,9,5,13)(3,7,11,14,12,15,10,16);(1,3,5,11)(2,7,8,14)(4,12,6,10)(9,15,13,16);(2,8)(7,14)(9,13)(15,16)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72209844, #72216336, #72216344, … |
| maximal normal subgroup of | #72209844, #72216336, #72216344, … |
| maximal quotient of | C2×#139731, #72218391, #72218855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46927);