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C₄×#1904 #354248

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,3,4);(5,6,9,12,10,14,8,13)(7,15,17,19,18,20,16,11);(5,7)(6,11,14,19)(8,16)(9,17)(10,18)(12,15,13,20);(6,12)(8,9)(11,15)(13,14)(16,17)(19,20)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#1904, C₄×#23882, #72226667, …
maximal normal subgroup of	C₄×#23882, #72226667, #72226671, …
maximal quotient of	AGL(1,𝔽₅)×#1904, #72257926, #72264005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261444);