Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
327 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 327 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(6,7)(8,9)(11,12)(13,14,15,18,16,19,20,17);(1,2,3,4)(5,6,9,11,10,12,8,7)(14,17)(15,20)(18,19);(5,7,8,12,10,11,9,6)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=eg,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136893, #136894, #136905, … |
| maximal normal subgroup of | #136893, #136894, #136905, … |
| maximal quotient of | #101996, #102020, #102218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2722);