Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = #72286229 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C42 |
| Frattini, prime radical, cosocle | #532 | C22 |
| socle | C2 | #1390 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 16 points:
(1,2,5,10,7,3,13,9)(4,14,6,11,15,16,8,12);(1,3,6,12,7,2,8,11)(4,9,5,14,15,10,13,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=df,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7492, #151812, #151825, … |
| maximal normal subgroup of | C2×#7492, #151812, #151825, … |
| maximal quotient of | #94089, #94132, #94145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 480);