Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3339 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,15,19)(11,16,14,30)(12,22,25,39)(13,26,29,38)(17,32,34,24)(18,21,36,28)(20,37,23,35)(27,31,40,33);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,19,30)(12,23,25,20)(13,21,29,28)(17,33,34,31)(18,26,36,38)(22,35,39,37)(24,40,32,27);(9,12,15,25)(10,17,19,34)(11,20,14,23)(13,27,29,40)(16,31,30,33)(18,35,36,37)(21,32,28,24)(22,38,39,26);(3,8)(5,7)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33887, #7623700, #7623716, … |
| maximal normal subgroup of | C2×#33887, #7623700, #7623716, … |
| maximal quotient of | #497419, #497563, #497714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26875);