Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) = #72216494 (order 1024) · Inn(G) = #106 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #106 |
| commutator | C8 | C2×C16 |
| Frattini, cosocle | C82 | C22 |
| socle, prime radical | C22 | #334 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,6,7,12)(3,9)(4,11)(8,15,14,13)(10,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2,3,6,4,8,9,13,5,7,10,12,11,14,16,15)Polycyclic presentation:
2^8|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,d^b=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93757, #93758, #93766, … |
| maximal normal subgroup of | #93757, #93758, #93766, … |
| maximal quotient of | #92913, #93876, #93877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 87);