Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(18,22)(19,23)(20,26)(24,32)(25,30)(27,31)(28,29);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9);(17,18,23,29,20,24,27,30,21,25,31,32,26,28,19,22)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eg,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #93872, #93873, #123303, … |
| maximal normal subgroup of | #93872, #93873, #123303, … |
| maximal quotient of | #135777, #141656, #141689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5509);