#9772

order	256
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=dfg,c^2=eh,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#131845, #136513, #136871, …
maximal normal subgroup of	#131845, #136513, #136871, …
maximal quotient of	#102036, #102046, #102220, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 2760);