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D₄×#354 #497301

order	512
nilpotency class	4
isoclinism	isoclinic class of #480872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄² ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1442 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)(7,13,15,20)(8,16,18,11)(9,12)(10,14)(17,19);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2127, #72263632, #72263636, …
maximal normal subgroup of	D₄×#2127, #72263632, #72263636, …
maximal quotient of	#72317692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404497);