Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) = #72282266 (order 2048) · Inn(G) = #327 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #327 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C22×M16 | C22 |
| socle | C2 | #1294 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 49 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 — 2 generators on 32 points:
(1,2,5,10,7,11,6,22)(3,12,14,27,16,29,15,30)(4,17,19,28,20,32,21,9)(8,23,24,31,26,18,25,13);(1,3)(2,8)(4,18,20,23)(5,14)(6,15)(7,16)(9,27,28,30)(10,24)(11,26)(12,17,29,32)(13,21,31,19)(22,25)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=h,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7256, #127587, #127614, … |
| maximal normal subgroup of | C2×#7256, #127587, #127614, … |
| maximal quotient of | #93035, #93037, #93043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 244);