Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | Aut(G) = #72234841 (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | #1867 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,6,11,7,12,5,19)(3,13,15,23,16,30,20,29)(4,10,17,9,18,26,21,27)(8,22,24,32,25,31,28,14);(1,3,6,15,7,16,5,20)(2,8,11,24,12,25,19,28)(4,14,17,22,18,32,21,31)(9,23,26,30,27,29,10,13);(1,4)(2,9)(3,14)(5,21)(6,17)(7,18)(8,23)(10,19)(11,26)(12,27)(13,28)(15,22)(16,32)(20,31)(24,30)(25,29)Polycyclic presentation:
2^8|a^2=f,b^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eh,g^a=gh,e^b=egh,d^c=dg,e^c=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152458, #152470, #152484, … |
| maximal normal subgroup of | #152458, #152470, #152484, … |
| maximal quotient of | #135938, #135939, #135960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5956);