Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72278713 (order 2048) · Inn(G) = #99 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C2×C16 |
| Frattini, cosocle | C2×C4⋊C8 | C22 |
| socle | C22 | #324 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,9,11,15)(4,12,13,7)(5,8)(6,10)(14,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,3)(2,6,8,14)(4,10)(5,11)(7,9,12,15)(13,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=g,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,d^b=deg,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #93689, #93690, #93698, … |
| maximal normal subgroup of | #93689, #93690, #93698, … |
| maximal quotient of | #92871, #92967, #93717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50);