Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C4×C8 |
| Frattini, cosocle | C22×M16 | C22 |
| socle | C22 | #324 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6,8,14,10,16,24,32,11,13,22,29,9,15,23,31)(7,17,18,30,20,27,34,35,21,12,25,26,19,28,33,36);(1,2,3,4)(5,7,9,19,11,21,10,20)(6,12,15,27,13,17,16,28)(8,18,23,33,22,25,24,34)(14,30,31,36,29,26,32,35)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,e^2=h,g^2=h,b^a=bc,c^a=cf,f^a=fh,d^b=df,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93900, #93902, #93914, … |
| maximal normal subgroup of | #93900, #93902, #93914, … |
| maximal quotient of | #92964, #92965, #92976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 180);