Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,23,33)(12,24,26,16)(13,27,20,17)(15,28,30,22)(21,35,37,32)(25,31,34,39)(29,36,38,40);(1,3,4,7)(2,5,6,8)(9,11,20,28,14,23,13,22)(10,15,27,33,18,30,17,19)(12,21,34,40,26,37,25,36)(16,29,39,35,24,38,31,32);(2,6)(5,8);(9,12,13,25,14,26,20,34)(10,16,17,31,18,24,27,39)(11,21,22,36,23,37,28,40)(15,29,19,32,30,38,33,35)Polycyclic presentation:
2^8|a^2=h,b^2=efh,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138013, #138014, #138288, … |
| maximal normal subgroup of | #138013, #138014, #138288, … |
| maximal quotient of | #351274, #351275, #351280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26112);