Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12090 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 200 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18)(19,23)(20,22)(21,24)(25,29)(26,31)(27,32)(28,30);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,22)(23,29)(24,30)(31,32);(17,19,21,26)(18,23,24,31)(20,27,28,25)(22,29,30,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=egh,d^c=dg,e^d=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151393, #151461, C2×#12215, … |
| maximal normal subgroup of | #151393, #151461, C2×#12215, … |
| maximal quotient of | #125969, #126010, #126079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5203);