Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#550 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
731 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 731 conjugacy classes of subgroups — too many to draw. Only its 201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,14,10,13,7,11)(8,16,15,20,17,12,18,19);(1,2)(3,4)(5,7,10,9)(6,11,13,14)(8,15,17,18)(12,19,16,20);(5,8,10,17)(6,12,13,16)(7,15,9,18)(11,19,14,20);(2,4)(5,7,10,9)(6,11,13,14)(8,15,17,18)(12,19,16,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #138888, #138890, #138957, … |
| maximal normal subgroup of | #138888, #138890, #138957, … |
| maximal quotient of | #219721, #219728, #219732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15400);