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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 360 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24);(1,2,6,8,4,7,3,5)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,23)(20,24);(2,7)(5,8)(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21);(10,15)(14,22)(16,23)(21,24)Polycyclic presentation:
2^8|b^2=eg,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 | #267901, #267903, #267933, … |
| maximal normal subgroup of | #267901, #267903, #267933, … |
| maximal quotient of | #189145, #197871, #197954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12422);