Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
421 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2,5,12,6,10,4,17)(3,13,14,27,15,8,20,24)(7,11,21,26,18,25,19,9)(16,29,31,28,30,23,32,22);(1,3,7,16)(2,8,11,23)(4,14,19,31)(5,20,21,32)(6,15,18,30)(9,22,17,24)(10,13,25,29)(12,27,26,28);(1,4,18,21)(2,9,25,12)(3,14,30,32)(5,7,19,6)(8,22,29,27)(10,26,11,17)(13,28,23,24)(15,20,16,31)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93127, #93128, #93131, … |
| maximal normal subgroup of | #93127, #93128, #93131, … |
| maximal quotient of | #97068, #97072, #97073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1544);