Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
205 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(10,13)(14,15)(17,18,20,22,21,24,19,23);(1,3,4,8)(2,5,6,7)(9,10,11,13)(12,14,16,15)(18,22)(19,20)(23,24);(10,14)(12,16)(13,15)(17,18,21,24)(19,22,20,23)Polycyclic presentation:
2^8|a^2=dh,b^2=deg,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10681, #243948, #243954, … |
| maximal normal subgroup of | C2×#10681, #243948, #243954, … |
| maximal quotient of | #105239, #117093, #117098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3669);