Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139743 |
| 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 | #23895 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
815 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 815 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)(3,9)(4,13,10,7)(5,8)(6,11,14,16)(12,15);(1,3,10,16)(2,6,13,9)(4,11,5,12)(7,15,8,14);(1,2)(3,6)(4,7)(5,8)(9,16)(10,13)(11,15)(12,14)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=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,e^d=ei,f^d=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | B(5,𝔽2), #72216336, #72216376, … |
| maximal normal subgroup of | B(5,𝔽2), #72216336, #72216376, … |
| maximal quotient of | #72216488, #72216717, C2×#139743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46939);