Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | Aut(G) = C2×#512862 (order 1024) · Inn(G) = #1402 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1402 |
| commutator | #127 | C2×C4 |
| Frattini, prime radical, cosocle | #493 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
81 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)(3,8,11,16)(4,12,13,15)(5,9)(6,10,7,14);(1,3,5,11)(2,6,9,12)(4,10,14,13)(7,8,15,16)Polycyclic presentation:
2^8|a^2=d,b^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,d^b=degh,e^b=egh,f^b=fgh,d^c=dfg,e^c=egh,f^c=fh,e^d=egh,f^d=fh22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7516, #153435, #153436, … |
| maximal normal subgroup of | C2×#7516, #153435, #153436, … |
| maximal quotient of | #94539, #94596, #94671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 504);