Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 103 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)(6,11,12,14)(8,16)(13,20,19,18)(15,17);(2,4)(5,6,8,13,9,14,17,20,10,12,16,19,7,11,15,18);(6,11,12,14)(8,16)(13,20,19,18)(15,17)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,f^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93550, #93551, #93639, … |
| maximal normal subgroup of | #93550, #93551, #93639, … |
| maximal quotient of | #128259, #128278, #128354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5593);