Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
188 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(2,6)(3,7)(4,8)(9,10,13,17,14,18,12,16)(11,19,21,15,22,23,20,24);(1,2,5,3)(4,7,8,6)(9,11,12,20,14,22,13,21)(10,15,16,19,18,24,17,23);(1,3,5,2)(4,6,8,7)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,24,23)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,d^c=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8909, #192958, #192960, … |
| maximal normal subgroup of | C2×#8909, #192958, #192960, … |
| maximal quotient of | #99396, #99398, #99400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1897);