Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 65536 |
| 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 | C24 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
731 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 731 conjugacy classes of subgroups — too many to draw. Only its 263 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)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(1,2,3,4)(8,16)(11,18)(14,20)(17,19);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138316, #138317, #138360, … |
| maximal normal subgroup of | #138316, #138317, #138360, … |
| maximal quotient of | #188275, #188276, #188278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13622);