Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12099 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| 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×C4⋊C4 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 197 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:
(1,2,6,12)(3,11,16,29)(4,17,19,10)(5,9,23,28)(7,14,21,13)(8,18,24,15)(20,30,32,27)(22,26,31,25);(1,3,6,16)(2,8,12,24)(4,18,19,15)(5,21,23,7)(9,27,28,30)(10,29,17,11)(13,25,14,26)(20,31,32,22);(2,9)(3,5)(4,19)(8,10)(11,14)(12,28)(13,29)(15,31)(16,23)(17,24)(18,22)(20,32)(25,30)(26,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,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^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12223, #484337, #484338, … |
| maximal normal subgroup of | C2×#12223, #484337, #484338, … |
| maximal quotient of | #125972, #125974, #126014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5211);