Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 147 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:
(2,3)(5,6,7,8)(9,10,13,16,14,17,18,11)(12,19,22,20,23,24,21,15);(1,2,4,3)(9,11,18,17,14,16,13,10)(12,20,21,19,23,15,22,24);(6,8)(9,12,14,23)(10,15,17,20)(11,19,16,24)(13,21,18,22);(1,3,4,2)(9,13,14,18)(10,16,17,11)(12,21,23,22)(15,24,20,19)Polycyclic presentation:
2^8|a^2=f,b^2=egh,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #501478, #501479, #501482, … |
| maximal normal subgroup of | #501478, #501479, #501482, … |
| maximal quotient of | #321253, #321261, #321270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21454);