Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #394 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,14,11,7,10,13)(8,16,18,15,19,20,17,12);(5,7)(6,11)(8,15)(9,13)(10,14)(12,19)(16,17)(18,20);(2,4)(5,8,9,17,11,19,10,18)(6,12,13,16,7,15,14,20)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11151, #357911, #358423, … |
| maximal normal subgroup of | C2×#11151, #357911, #358423, … |
| maximal quotient of | #118666, #118699, #118700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4139);