Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,22)(12,28)(13,31)(14,20)(15,21)(16,25)(17,24)(18,33)(19,32)(23,38)(26,36)(27,39)(29,35)(30,40)(34,37);(2,5)(6,7)(9,11,14,25,15,27,32,26)(10,16,19,22,21,36,20,39)(12,13,24,33,30,34,29,23)(17,37,28,18,35,31,40,38);(9,12,14,24,15,30,32,29)(10,17,19,28,21,35,20,40)(11,23,25,13,27,33,26,34)(16,18,22,31,36,38,39,37);(1,3,4,8)(2,6,5,7)(9,13,32,23,15,34,14,33)(10,18,20,37,21,38,19,31)(11,24,26,12,27,29,25,30)(16,35,39,28,36,17,22,40)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #503906, #503907, #503910, … |
| maximal normal subgroup of | #503906, #503907, #503910, … |
| maximal quotient of | #307339, #307342, #307346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19259);