Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#127 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
367 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(18,22)(23,30)(24,29)(25,31);(3,11)(7,15)(10,16)(12,13)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)Polycyclic presentation:
2^8|b^2=eh,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg52 conjugacy classes — view the full character table.
| maximal subgroup of | #143177, #143179, #143183, … |
| maximal normal subgroup of | #143177, #143179, #143183, … |
| maximal quotient of | #191428, #206092, #206096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12601);