Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#459 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,20,19);(2,4)(13,15,16,20)(14,18,17,19);(6,9)(10,12)(14,17)(15,19)(18,20);(2,4)(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=eh,b^2=h,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26597, #6439397, #6439417, … |
| maximal normal subgroup of | C2×#26597, #6439397, #6439417, … |
| maximal quotient of | #310139, #310140, #310143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19585);