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D₄×#365 #497835

order	512
nilpotency class	4
isoclinism	isoclinic class of #483463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

662 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);(6,10,11,16)(7,13)(8,15)(12,19,18,17)(14,20);(5,6,7,10,8,12,13,17,9,11,14,16,15,18,20,19)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227119, D₄×AGL(1,ℤ/16), #72227173, …
maximal normal subgroup of	#72227119, D₄×AGL(1,ℤ/16), #72227173, …
maximal quotient of	D₄×#1368, #72227118, #72227119, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405031);