Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| 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 | |
322 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(1,2,5,7)(3,8,6,4)(10,14)(11,17)(12,20)(16,22)(18,19)(21,24);(1,3,4,2,5,6,8,7)(9,10,12,14,13,16,20,22)(11,15,18,21,19,23,17,24);(9,11,13,19)(10,15,16,23)(12,18,20,17)(14,21,22,24);(10,16)(14,22)(15,23)(21,24)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,d^b=dh,f^c=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #267862, #267864, #267894, … |
| maximal normal subgroup of | #267862, #267864, #267894, … |
| maximal quotient of | #189135, #197857, #197934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12391);