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 | #3285 | 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)(9,10,14,18)(11,19,20,37)(12,23,27,38)(13,28,25,17)(15,30,31,21)(16,33,35,26)(22,39,29,34)(24,36,40,32);(1,3)(2,5)(4,8)(6,7)(11,20)(15,31)(19,37)(21,30)(22,29)(24,40)(32,36)(34,39);(9,11,14,20)(10,15,18,31)(12,24,27,40)(13,29,25,22)(16,34,35,39)(17,36,28,32)(19,38,37,23)(21,33,30,26);(3,8)(5,7)Polycyclic presentation:
2^8|a^2=eh,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×#33883, #7623700, #7623717, … |
| maximal normal subgroup of | C2×#33883, #7623700, #7623717, … |
| maximal quotient of | #497415, #497567, #497714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26871);