Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #2021 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2021 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3021 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 3 generators on 32 points:
(1,2,6,11)(3,12,16,27)(4,17,21,9)(5,8,22,10)(7,18,25,14)(13,30,15,29)(19,28,20,26)(23,31,24,32);(1,3)(2,7)(4,18)(5,13)(6,16)(8,24)(9,27)(10,23)(11,25)(12,17)(14,21)(15,22)(19,32)(20,31)(26,29)(28,30);(2,8)(3,13)(5,22)(9,26)(10,11)(12,27)(14,32)(15,16)(17,28)(18,31)(19,20)(23,24)Polycyclic presentation:
2^8|a^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,f^b=fh,g^b=gh,d^c=dg,e^c=eh,f^c=fh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13361, #498915, #498916, … |
| maximal normal subgroup of | C2×#13361, #498915, #498916, … |
| maximal quotient of | #142229, #142230, #142276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6349);