#19479

order	256
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,6,8)(3,4,7,5)(9,10,14,18)(11,15,20,29)(12,22,24,32)(13,25,27,17)(16,26,31,23)(19,33,34,40)(21,37,36,30)(28,38,39,35);(1,3,5,2,6,7,4,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,30)(18,29)(22,33)(23,35)(24,34)(25,37)(26,38)(27,36)(31,39)(32,40);(9,12,14,24)(10,16,18,31)(11,19,20,34)(13,26,27,23)(15,28,29,39)(17,32,25,22)(21,38,36,35)(30,40,37,33);(1,4,6,5)(2,7,8,3)(11,20)(15,29)(19,34)(21,36)(28,39)(30,37)(33,40)(35,38)

Polycyclic presentation:

2^8|a^2=gh,b^2=e,c^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,f^c=fh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19479, #6354548, #6354580, …
maximal normal subgroup of	C₂×#19479, #6354548, #6354580, …
maximal quotient of	#189155, #197887, #197966, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12467);