Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = C23×#2196 (order 1024) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C8 |
| Frattini, cosocle | #391 | C22 |
| prime radical | C2×2−1+4 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10,11,15)(12,21,22,18)(13,17,20,24)(14,23,16,19);(9,10)(11,18)(12,21)(13,17)(14,20)(15,19)(16,23)(22,24)Polycyclic presentation:
2^8|a^2=d,c^2=h,d^2=f,e^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ch,d^b=de,e^b=eh,g^b=gh,d^c=dgh,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7345, #149542, #149543, … |
| maximal normal subgroup of | C2×#7345, #149542, #149543, … |
| maximal quotient of | #93192, #93193, #93198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 333);