Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,25,27,32)(21,29,26,24)(23,31,30,28);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,22)(23,30)(24,29)(25,32);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,26,31,20,27,21,28)(18,23,29,32,22,30,24,25);(1,4,5,3)(2,7,8,6)(18,22)(19,27)(24,29)(28,31)Polycyclic presentation:
2^8|a^2=eg,b^2=egh,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100919, #297062, #297065, … |
| maximal normal subgroup of | #100919, #297062, #297065, … |
| maximal quotient of | #362287, #362929, #365669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22215);