Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| automorphism | Aut(G) = #72279473 (order 2048) · Inn(G) = #1629 · Out(G) = C22⋊C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1629 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3293 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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:
(3,11,13,30)(4,15)(5,18)(7,22,24,32)(8,26)(9,21)(12,17,14,29)(16,28)(19,20)(23,27,25,31);(1,2,6,10)(3,7,13,24)(4,16,20,26)(5,21,18,9)(8,15,28,19)(11,27,30,31)(12,25,14,23)(17,22,29,32);(1,3)(2,7)(4,17)(5,12)(6,13)(8,27)(9,23)(10,24)(11,19)(14,18)(15,30)(16,22)(20,29)(21,25)(26,32)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,g^a=gh,d^b=df,e^b=eg,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #153645, #153646, #153647, … |
| maximal normal subgroup of | #153645, #153646, #153647, … |
| maximal quotient of | #125893, #125895, #125948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5355);