Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) = #72313686 (order 4096) · Inn(G) = #447 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1852 |
| prime radical | C24 | Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
193 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,5,10,7,11,6,21)(3,12,15,29,14,28,22,26)(4,16,19,27,18,31,20,9)(8,17,24,30,23,32,25,13);(1,3,5,15,7,14,6,22)(2,8,10,24,11,23,21,25)(4,17,19,30,18,32,20,13)(9,26,16,12,27,29,31,28);(4,18)(9,27)(12,28)(13,30)(16,31)(17,32)(19,20)(26,29)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #135992, #135995, #136048, … |
| maximal normal subgroup of | #135992, #135995, #136048, … |
| maximal quotient of | #135816, #135842, #135985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5912);