Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| 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 | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,4,5)(3,7,8,6)(9,10,16,22)(11,23,28,36)(12,29,32,39)(13,33,30,38)(14,21,15,20)(17,27,37,26)(18,25,40,31)(19,24,35,34);(2,5)(6,7)(9,11,15,27,16,28,14,26)(10,17,21,36,22,37,20,23)(12,30,31,34,32,13,25,24)(18,38,29,35,40,33,39,19);(9,12,15,31,16,32,14,25)(10,18,21,29,22,40,20,39)(11,24,27,30,28,34,26,13)(17,35,36,33,37,19,23,38);(1,3,4,8)(2,6,5,7)(9,13,15,24,16,30,14,34)(10,19,21,38,22,35,20,33)(11,25,27,12,28,31,26,32)(17,18,36,29,37,40,23,39)Polycyclic presentation:
2^8|a^2=efh,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,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #504175, #504799, #510294, … |
| maximal normal subgroup of | #504175, #504799, #510294, … |
| maximal quotient of | #308988, #308991, #308995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19493);