Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(2,5)(3,8)(6,7)(11,18)(14,22)(15,20)(19,24);(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24)Polycyclic presentation:
2^8|c^2=gh,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=egh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151162, #151163, #151248, … |
| maximal normal subgroup of | #151162, #151163, #151248, … |
| maximal quotient of | #123155, #123156, #123157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5144);