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C₄×#1790 #354783

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸×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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#32898, AGL(1,𝔽₅)×#1790, C₄×#24260, …
maximal normal subgroup of	C₄×#32898, C₄×#24260, #72264504, …
maximal quotient of	AGL(1,𝔽₅)×#1790, C₄×#12753, C₄×#8534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261979);