Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #486 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #486 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2023 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 182 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,14,29)(4,17)(5,10)(6,22)(7,11)(8,15,24,31)(9,21)(13,30,16,26)(18,28)(19,27,32,23)(20,25);(3,13,14,16)(4,18)(8,23,24,27)(9,25)(12,26,29,30)(15,32,31,19)(17,28)(20,21);(1,3)(2,8)(4,19)(5,16)(6,13)(7,14)(9,26)(10,27)(11,24)(12,28)(15,20)(17,29)(18,32)(21,31)(22,23)(25,30)Polycyclic presentation:
2^8|a^2=d,b^2=d,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^b=eh,d^c=dh,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125888, #125889, #126214, … |
| maximal normal subgroup of | #125888, #125889, #126214, … |
| maximal quotient of | #144452, #144453, #144482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6361);