Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 108 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(18,22)(19,23)(20,26)(24,28)(25,32)(27,31)(29,30);(2,6)(7,15)(8,13)(9,16)(17,18,23,30,20,25,27,28,21,24,31,32,26,29,19,22);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(18,24)(22,28)(25,29)(30,32)Polycyclic presentation:
2^8|b^2=dfg,c^2=g,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,c^b=ch,e^c=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12479, #490629, #490630, … |
| maximal normal subgroup of | C2×#12479, #490629, #490630, … |
| maximal quotient of | #135011, #138593, #138671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5467);