Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
265 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(1,2,6,8,5,3,4,7)(10,14,17,20)(11,18,19,21);(1,3,5,2)(4,7,6,8)(9,10,15,14,13,17,12,20)(11,16,18,22,19,24,21,23);(4,6)(7,8)(9,11)(10,16)(12,21)(13,19)(14,22)(15,18)(17,24)(20,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #94776, #94778, #94780, … |
| maximal normal subgroup of | #94776, #94778, #94780, … |
| maximal quotient of | #99599, #99601, #99602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2057);