Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3497 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
549 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 117 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 — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,16)(11,18,19,21)(12,20,15,23)(13,22,24,17);(2,6)(3,4)(7,8)(9,11)(10,15)(12,16)(13,23)(14,19)(17,21)(18,22)(20,24);(9,12,14,15)(10,11,16,19)(13,18,24,21)(17,23,22,20);(1,3,5,4)(2,7,8,6)(10,16)(11,19)(17,22)(20,23)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244588, #244589, #244591, … |
| maximal normal subgroup of | #244588, #244589, #244591, … |
| maximal quotient of | #168961, #169009, #169073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11850);