Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
517 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,5,8)(3,7,4,6)(9,10)(11,19)(12,14)(13,17)(15,18)(16,20)(21,24)(22,23);(1,3,5,4)(2,6,8,7)(9,11,13,20)(10,16,17,19)(12,22,18,24)(14,21,15,23);(1,3,5,4)(2,7,8,6)(9,12,15,17,13,18,14,10)(11,16,23,22,20,19,21,24);(1,4,5,3)(2,7,8,6)(10,17)(12,18)(16,19)(22,24)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=ef,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268293, #268298, #268321, … |
| maximal normal subgroup of | #268293, #268298, #268321, … |
| maximal quotient of | #304370, #304372, #304378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18916);