Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72283959 (order 1024) · Inn(G) = #99 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator | C2×C4 | C2×C16 |
| Frattini, cosocle | C2×M32 | C22 |
| socle | C2 | #1322 |
| prime radical | C23 | M32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,5,12,17,25,6,9,16,22,18,23,29,31,7,11,15,24,19,26,28,32,3,8,13,20,14,21,27,30);(2,8,11,9)(4,15)(10,20,24,22)(12,21,26,23)(13,16)(17,28)(25,30,32,31)(27,29)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh,d^b=deh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93693, #93694, #93699, … |
| maximal normal subgroup of | #93693, #93694, #93699, … |
| maximal quotient of | #93717, #93723, #93731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 352);