Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12416 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 111 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18)(19,22)(20,23)(21,29)(24,28)(25,27)(26,30)(31,32);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(18,22)(23,30)(24,31)(25,29);(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,e^c=eg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12449, #490493, #490494, … |
| maximal normal subgroup of | C2×#12449, #490493, #490494, … |
| maximal quotient of | #134952, #138607, #138609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5437);