Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
244 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,14,18,23)(12,15)(13,16)(17,20)(19,21,24,22);(1,2,3,5,4,6,7,8)(9,11)(10,14,16,23)(12,19)(13,18)(15,21,17,22)(20,24);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9303, #199168, #199169, … |
| maximal normal subgroup of | C2×#9303, #199168, #199169, … |
| maximal quotient of | #99963, #99967, #99979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2291);