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) = C22 |
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 | |
194 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 194 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13)(3,16,20,32)(4,21,23,30)(5,25,18,11)(6,12,26,28)(8,15,17,14)(9,24,29,19)(10,27,31,22);(1,3,17,26)(2,9,14,21)(4,18,19,27)(5,24,22,23)(6,7,20,8)(10,28,25,16)(11,32,31,12)(13,29,15,30);(1,4,8,24)(2,10,15,11)(3,18,6,22)(5,26,27,20)(7,23,17,19)(9,28,30,32)(12,21,16,29)(13,31,14,25)Polycyclic presentation:
2^8|a^2=f,b^2=fh,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, #151460, C2×#12216, … |
| maximal normal subgroup of | #151393, #151460, C2×#12216, … |
| maximal quotient of | #125971, #126012, #126080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5204);