Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = #72278713 (order 2048) · Inn(G) = #99 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C22 |
| socle | C22 | #323 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
177 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,19,28,36,15,20,29,37,12,17,26,34)(11,21,23,38,22,32,39,35,24,33,27,40,25,16,30,31);(2,5)(6,7)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,33)(18,35)(19,21)(20,32)(23,29)(26,39)(28,30)(31,37)(34,38)(36,40)Polycyclic presentation:
2^8|a^2=d,c^2=h,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=ch,d^b=deh,d^c=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | #93202, #93204, #93205, … |
| maximal normal subgroup of | #93202, #93204, #93205, … |
| maximal quotient of | #92815, #92886, #92987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46);