Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) = C4×#21452 (order 1024) · Inn(G) = #102 · Out(G) = C2×C42 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C2×C8 |
| Frattini, cosocle | C82 | C22 |
| socle, prime radical | C22 | D4⋊C8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
67 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)(17,18,19,23,21,24,26,30,20,22,25,28,27,29,31,32);(1,2,5,7)(3,10,11,15)(4,13,12,8)(6,9,14,16)(18,22)(19,25)(24,29)(26,31)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=eh,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cegh,d^b=dgh,e^b=eg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7075, #122512, #122513, … |
| maximal normal subgroup of | C2×#7075, #122512, #122513, … |
| maximal quotient of | #92822, #92869, #92962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 63);