Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | Aut(G) = C2×#72263997 (order 2048) · Inn(G) = #327 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #327 |
| commutator | C22×C8 | C2×C4 |
| Frattini, cosocle | C2×C4⋊C8 | C22 |
| prime radical | C22×D8 | C4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
120 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,19,22,14)(12,16)(13,18)(15,20,23,24)(17,21);(2,5)(10,14)(11,20)(12,21)(15,17)(16,23)(18,19)(22,24)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=g,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fh,d^c=dgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7080, #122576, #122580, … |
| maximal normal subgroup of | C2×#7080, #122576, #122580, … |
| maximal quotient of | #92845, #92881, #93033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 68);