Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7535 |
| automorphism | Aut(G) = #72286283 (order 2048) · Inn(G) = #1409 · Out(G) = C22⋊C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1409 |
| commutator | C4×C8 | C2×C4 |
| Frattini, cosocle | #492 | C22 |
| prime radical | #127 | D4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
75 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)(3,10,11,6)(4,15,12,9)(5,8)(7,13,16,14);(2,6,8,16)(3,11)(4,12)(7,9,10,15)(13,14)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=fgh,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfgh,d^b=defh,e^b=egh,f^b=fh,d^c=dfg,e^d=egh,f^d=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7535, #153533, #153534, … |
| maximal normal subgroup of | C2×#7535, #153533, #153534, … |
| maximal quotient of | #94572, #94590, #94654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 523);