Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,18,15,19,12,16)(11,20,22,13,17,21,24,23);(2,5)(6,7)(9,11,21,18,15,17,20,16)(10,14,24,13,19,12,22,23);(1,3,4,8)(2,6,5,7)(9,12,15,14)(10,16,19,18)(11,22,17,24)(13,21,23,20)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=eh,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dgh,f^b=fgh,g^b=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132417, #132418, #132419, … |
| maximal normal subgroup of | #132417, #132418, #132419, … |
| maximal quotient of | #141939, #141948, #141959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6291);