Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(2,4)(5,6,9,15,10,14,17,20,11,13,18,12,8,19,7,16);(1,2,3,4)(5,7,8,18,11,17,10,9)(6,12,14,16,13,15,19,20);(6,13)(7,17)(9,18)(14,19)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ch,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93360, #124440, #124937, … |
| maximal normal subgroup of | #93360, #124440, #124937, … |
| maximal quotient of | #134223, #134251, #134276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5779);