Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3334 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(1,3)(2,6)(4,7)(5,8);(1,4,5,2)(3,7,8,6)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(3,8)(6,7)Polycyclic presentation:
2^8|a^2=eh,c^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=de,f^c=fg,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #491905, #491906, #491912, … |
| maximal normal subgroup of | #491905, #491906, #491912, … |
| maximal quotient of | #497538, #497540, #497772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26890);