Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 518 conjugacy classes of subgroups — too many to draw. Only its 137 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)(9,10,14,18)(11,15,21,24)(12,16,13,17)(19,22,20,23);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,20)(13,19)(14,21)(16,23)(17,22)(18,24);(10,16)(11,19)(12,13)(17,18)(20,21)(22,23);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=eh,b^2=e,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,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #250103, #250109, #250112, … |
| maximal normal subgroup of | #250103, #250109, #250112, … |
| maximal quotient of | #265644, #265645, #265656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17309);