Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,11,15)(7,16,13,18)(8,17,12,9)(10,20,19,14);(1,2)(3,4)(5,7,11,13)(6,12,15,8)(9,19,17,10)(14,18,20,16);(2,4)(5,8)(6,13)(7,15)(9,14)(10,18)(11,12)(16,19)(17,20);(5,9,10,7,11,17,19,13)(6,8,14,18,15,12,20,16)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,e^b=eh,f^c=fh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151260, #151262, #151428, … |
| maximal normal subgroup of | #151260, #151262, #151428, … |
| maximal quotient of | #339711, #339713, #339727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23857);