Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = C22×C4×B(4,𝔽2) (order 1024) · Inn(G) = #99 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C22 |
| socle | C22 | #324 |
| prime radical | C24 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,13,16,21,23,14,18,20,24,11,15,19,22);(2,5,6,7)(3,8)(10,15,18,16)(12,20)(17,22,24,23)(19,21)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cgh,d^b=deh,d^c=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7067, #122345, #122346, … |
| maximal normal subgroup of | C2×#7067, #122345, #122346, … |
| maximal quotient of | #92871, #92872, #92989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55);