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D₄×#347 #493224

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

736 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);(2,4);(5,6,7,11,8,13,15,19,9,12,14,20,16,10,17,18);(6,10,12,13)(7,14)(11,19,20,18)(15,17)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225552, D₄×AGL(1,ℤ/16), #72227176, …
maximal normal subgroup of	#72225552, D₄×AGL(1,ℤ/16), #72227176, …
maximal quotient of	D₄×#1313, #72225551, #72225552, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400420);