Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = #72234841 (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1790 |
| prime radical | #3529 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,9,6,10,21,28)(3,11,14,29,12,25,19,31)(4,15,18,27,17,8,20,26)(7,22,24,16,23,30,32,13);(1,3)(2,7)(4,16)(5,19)(6,12)(8,25)(9,24)(10,23)(11,15)(13,17)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32);(3,12)(7,23)(8,15)(13,16)(14,19)(22,30)(24,32)(26,27)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,d^b=dh,e^b=egh,f^b=fgh,d^c=dg,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152553, #152579, #152612, … |
| maximal normal subgroup of | #152553, #152579, #152612, … |
| maximal quotient of | #129610, #129611, #129612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5754);