Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12298 |
| automorphism | Aut(G) = #72308157 (order 2048) · Inn(G) = #1613 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,19,6,10,21,9)(3,11,14,24,13,27,20,28)(4,15,18,8,17,31,32,25)(7,16,26,12,23,29,22,30);(1,3,6,13)(2,7,10,23)(4,16,17,29)(5,20,21,14)(8,24,25,28)(9,26,19,22)(11,15,27,31)(12,18,30,32);(8,25)(11,27)(15,31)(24,28)Polycyclic presentation:
2^8|a^2=fg,b^2=h,d^2=f,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151202, #151203, #151206, … |
| maximal normal subgroup of | #151202, #151203, #151206, … |
| maximal quotient of | #126285, #126442, #126446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5287);