Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#389 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
422 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,21)(3,12,14,8,16,23,15,24)(4,17,20,25,18,31,19,9)(13,22,28,30,27,32,29,26);(1,3,6,14,7,16,5,15)(2,8,10,23,11,24,21,12)(4,13,20,28,18,27,19,29)(9,22,17,30,25,32,31,26);(1,4,5,19,7,18,6,20)(2,9,10,17,11,25,21,31)(3,13,15,29,16,27,14,28)(8,22,23,30,24,32,12,26);(4,18)(9,25)(13,27)(17,31)(19,20)(22,32)(26,30)(28,29)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #367511, #367550, #370116, … |
| maximal normal subgroup of | #367511, #367550, #370116, … |
| maximal quotient of | #203711, #203715, #203719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14172);