Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19420 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
312 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,5,6)(3,8,7,4)(10,14)(11,17)(12,21)(16,23)(19,20)(22,24);(1,3,4,2,5,7,8,6)(9,10,12,14,13,16,21,23)(11,18,19,24,20,15,17,22);(2,6)(3,7)(9,11,13,20)(10,15,16,18)(12,19,21,17)(14,22,23,24);(10,16)(14,23)(15,18)(22,24)Polycyclic presentation:
2^8|a^2=gh,b^2=e,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #267910, #267912, #267942, … |
| maximal normal subgroup of | #267910, #267912, #267942, … |
| maximal quotient of | #189144, #197871, #197950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12427);