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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #393 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 333 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:
(5,6)(7,14)(8,16,18,15)(9,12)(10,13)(11,17,19,20);(1,2)(3,4)(5,7,9,6,10,12,14,13)(8,16,17,19,18,15,20,11);(2,4)(5,8,9,17,10,18,14,20)(6,11,7,15,13,19,12,16)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,e^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×#11263, #358276, #358423, … |
| maximal normal subgroup of | C2×#11263, #358276, #358423, … |
| maximal quotient of | #120692, #120693, #120710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4251);