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C₄×#1889 #354283

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

(1,2,3,4);(6,8)(9,10)(11,15,16,13)(12,14);(5,6,7,8)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,16)(14,15)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#23900, AGL(1,𝔽₅)×#1889, C₄×#12753, …
maximal normal subgroup of	C₄×#23900, C₄×#12753, C₄×#23996, …
maximal quotient of	AGL(1,𝔽₅)×#1889, C₄×#9495, #72264015, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261479);