Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D8 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
313 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(2,5)(3,8)(6,7)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,2)(3,7)(4,6)(5,8);(2,6)(5,7)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(2,7)(3,8)(5,6)Polycyclic presentation:
2^8|a^2=h,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=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #491720, #491722, #491736, … |
| maximal normal subgroup of | #491720, #491722, #491736, … |
| maximal quotient of | #220780, #220956, #220976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12914);