Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = C2×C4×#3109 (order 1024) · Inn(G) = #102 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C2×C16 |
| Frattini, cosocle | C82 | C22 |
| socle, prime radical | C22 | D4⋊C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,6,7,14)(3,9)(4,12)(8,10,13,15)(11,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2,5,7)(3,10,11,15)(4,13,12,8)(6,9,14,16)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=e,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=ceg,d^b=dg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7071, #122464, #122465, … |
| maximal normal subgroup of | C2×#7071, #122464, #122465, … |
| maximal quotient of | #92822, #92869, #92942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 59);