Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | Aut(G) order 8192 |
| 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 | #466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 248 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:
(2,5)(3,7)(9,10,14,19)(11,15,22,30)(12,16,25,32)(13,26,24,18)(17,33,31,23)(20,27,36,39)(21,37,35,29)(28,40,38,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,19,30)(12,20,25,36)(13,21,24,35)(16,27,32,39)(17,28,31,38)(18,29,26,37)(23,34,33,40);(2,5)(6,8)(10,16)(12,23)(13,24)(15,27)(17,18)(19,32)(20,34)(21,35)(25,33)(26,31)(28,29)(30,39)(36,40)(37,38);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #489553, #489556, #489591, … |
| maximal normal subgroup of | #489553, #489556, #489591, … |
| maximal quotient of | #496587, #496589, #496700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26802);