Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10238 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3036 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
305 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,23)(19,24)(21,22);(10,14)(12,20)(15,23)(16,17)(18,21)(22,24);(2,5)(3,8)(6,7)(9,11,13,19)(10,15,17,24)(12,21,20,18)(14,22,16,23)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10239, #237551, #237569, … |
| maximal normal subgroup of | C2×#10239, #237551, #237569, … |
| maximal quotient of | #102507, #102509, #102510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3227);