#23200

order	256
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂³	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#136489, #136513, #136538, …
maximal normal subgroup of	#136489, #136513, #136538, …
maximal quotient of	#231468, #231474, #231482, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16188);