Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| 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 | #3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
611 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 611 conjugacy classes of subgroups — too many to draw. Only its 131 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,20,23)(12,21,19,24)(13,17,22,16);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,22)(13,19)(14,20)(16,21)(17,24)(18,23);(1,3,5,4)(2,7,8,6)(9,12,13,11,14,19,22,20)(10,15,16,21,18,23,17,24);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #244674, #244675, #244676, … |
| maximal normal subgroup of | #244674, #244675, #244676, … |
| maximal quotient of | #303206, #303207, #303210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18864);