Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) = #72313686 (order 4096) · Inn(G) = #481 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #481 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1853 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 201 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:
(1,2,5,9,6,11,21,10)(3,12,15,30,14,29,20,26)(4,16,19,8,18,28,31,27)(7,22,24,13,23,17,25,32);(1,3,6,14)(2,7)(4,17,18,22)(5,20,21,15)(8,26)(9,25)(10,24)(11,23)(12,16)(13,31,32,19)(27,30)(28,29);(4,18)(8,27)(12,29)(13,32)(16,28)(17,22)(19,31)(26,30)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #136014, #136017, #136067, … |
| maximal normal subgroup of | #136014, #136017, #136067, … |
| maximal quotient of | #135828, #135847, #136009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5920);