Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,9)(7,11,12,10)(14,18)(15,19)(16,20);(1,2)(3,4)(6,9)(7,12)(13,14,17,18)(15,19,20,16);(2,4)(13,15,17,20)(14,19,18,16);(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632670, #6632679, #6632686, … |
| maximal normal subgroup of | #6632670, #6632679, #6632686, … |
| maximal quotient of | #340768, #340770, #340776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23925);