Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C25×C4×D4 (order 1024) · Inn(G) = C2×D4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C2×D4 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
112 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 112 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,16,13,11,18,19,14,17,21,24,22,20,15,23);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,12,19,13,17,18,24,14,20,21,23,22,10,15,16)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=cfh,d^a=df,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12057, #483558, #483561, … |
| maximal normal subgroup of | C2×#12057, #483558, #483561, … |
| maximal quotient of | #128683, #128699, #129169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5045);