Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | Aut(G) = C22×D4×B(4,𝔽2) (order 2048) · Inn(G) = #342 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×C4×D4 | C22 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
186 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,6)(3,4)(7,8)(9,10)(11,15,16,13)(12,14);(1,2,4,7,5,8,3,6)(10,13)(14,15)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,e^2=fg,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7236, #127352, #127353, … |
| maximal normal subgroup of | C2×#7236, #127352, #127353, … |
| maximal quotient of | #93011, #93012, #93017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 224);