Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 4096 |
| 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 | |
299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,24)(13,29)(14,19)(15,20)(17,26)(18,22)(21,35)(23,33)(25,31)(27,36)(28,37)(30,32)(34,38)(39,40);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,29)(19,32)(20,33)(21,27)(24,31)(26,38)(28,39)(35,36)(37,40);(9,12,15,28)(10,17,20,36)(11,21,23,38)(13,27,14,26)(16,31,33,40)(18,35,32,34)(19,37,29,24)(22,39,30,25);(1,4,5,6)(2,3,7,8)(9,11)(10,18)(12,21)(13,30)(14,22)(15,23)(16,29)(17,35)(19,33)(20,32)(24,31)(25,27)(26,39)(28,38)(34,36)(37,40)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6672288, #6672289, #6672294, … |
| maximal normal subgroup of | #6672288, #6672289, #6672294, … |
| maximal quotient of | #341454, #341455, #341456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23968);