Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| 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 | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
641 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 641 conjugacy classes of subgroups — too many to draw. Only its 119 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,18)(14,19)(15,20)(17,23)(21,24);(2,5)(6,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,24)(19,23);(1,3)(2,6)(4,7)(5,8)(9,12,15,16)(10,17,20,21)(11,19,22,18)(13,23,14,24);(3,7)(6,8)(9,13,15,14)(10,18,20,19)(11,21,22,17)(12,23,16,24)Polycyclic presentation:
2^8|c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,g^b=gh,d^c=de,f^c=fh,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246377, #246378, #246379, … |
| maximal normal subgroup of | #246377, #246378, #246379, … |
| maximal quotient of | #326690, #326698, #326706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22609);