Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = #72287397 (order 4096) · Inn(G) = #354 · Out(G) = D42 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #354 |
| commutator | C42 | C42 |
| Frattini, prime radical, cosocle | #532 | C22 |
| socle | C2 | #1393 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,8,5,9,11,16)(4,7,10,15,13,6,12,14);(2,6,9,7)(3,10,11,12)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=g,d^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,d^b=dfgh,d^c=dgh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94640, #94641, #94652, … |
| maximal normal subgroup of | #94640, #94641, #94652, … |
| maximal quotient of | #94102, #94155, #94178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 489);